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# Contents:  VertexEnum.m VertexEnum-notebook.ma Power.m FaceLattice.m
#   FaceLattice-notebook.ma
# Wrapped by fukuda@sma on Tue Nov 24 08:59:02 1992
PATH=/bin:/usr/bin:/usr/ucb ; export PATH
if test -f 'VertexEnum.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'VertexEnum.m'\"
else
echo shar: Extracting \"'VertexEnum.m'\" \(31187 characters\)
sed "s/^X//" >'VertexEnum.m' <<'END_OF_FILE'
X(*
X
X  Vertex Enumeration for Convex Polytopes
X             and Arrangements
X
X             Version 0.41 Beta
X              November 14, 1992
X
X           Copyright (c) 1991,1992 by
X    Komei Fukuda and Ichiro Mizukoshi
X
XThis package contains Mathematica implementations
Xof Avis-Fukuda algorithms for enumerating all 
Xvertices of a convex polyhedra given by a system
Xof linear inequalities, and all vertices (points)
Xof an arrangement given similarly.
X
XThis package is copyright 1991 by Komei Fukuda and
XIchiro Mizukoshi.  This package may be copied in 
Xits entirety for nonprofit purposes only.  
XSale, other than for the direct cost of the media,
Xis prohibited.  This copyright notice must
Xaccompany all copies.
X
XThe authors make no representations, express or 
Ximplied, with respond to this documentation, of 
Xthe software it describes and contains, including
Xwithout limitations, any implied warranties of 
Xmechantability or fitness for a particular purpose,
Xall of which are expressly disclaimed.  The authors
Xshall in no event be liable for any indirect,
Xincidental, or consequential damages.
X
XThis beta release is designed to run under 
XVersion 1.2 & 2.0 of Mathematica. Any comments, 
Xbug reports, or requests to get on the 
XVertexEnumeration mailing list should be 
Xforwarded to:
X
X  Komei Fukuda  or Ichiro Mizukoshi
X  Graduate School of Systems Management
X  University of Tsukuba, Tokyo
X  3-29-1 Otsuka, Bunkyo-ku
X  Tokyo 112, Japan
X
X  +81-3-3942-6876
X  fukuda@gssm.otsuka.tsukuba.ac.jp
X  or mizukosi@gssm.otsuka.tsukuba.ac.jp
X 
X*)
X
X
XBeginPackage["VertexEnum`"]
X
XUnprotect[VE]
XUnprotect[PE]
XUnprotect[VertexEnumeration]
XUnprotect[PointEnumeration]
XUnprotect[CrissCrossSolve]
XUnprotect[BlandSolve]
XUnprotect[Faces]
XUnprotect[Polyhedron3D]
XUnprotect[SubsetQ]
XUnprotect[AdjacentQ]
XUnprotect[EdgesOfPolyhedron]
XUnprotect[MaximalQ]
XUnprotect[MaximalSets]
XUnprotect[LinesOfArrangement]
X
XVE::usage="VE[m,b] is a short form of VertexEnumeration[m,b].";
X
XPE::usage="PE[m,b] is a short form of PointEnumeration[m,b].";
X
XVertexEnumeration::usage="VertexEnumeration[m,b] gives the list of all vertices of the polyhedron represented as the solution set of the linear inequality system  m.x <= b, x>=0. If you want to get results on progress MonitoringFile -> \"stdout\" ";
X
XPointEnumeration::usage="PointEnumeration[m,b]  gives the list of all vertices of the arrangement of hyperplanes associated with the linear inequality system m.x <= b, x>=0.";
X
XCrissCrossSolve::usage="CrissCrossSolve[c,m,b] solves the linear program  maximize c.x subject to m.x<=b and x>=0 by the Criss-Cross method. The output is {x*, v*, y*} where x* is an optimal solution, v* is an optimal value, and y* is a dual optimal solution. It warns if it is infeasible or dualinfeasible";
X
XBlandSolve::usage="BlandSolve[c,m,b] solves the linear program  maximize c.x subject to m.x<=b and x>=0 by the Simplex method with Bland's rule, provided b is nonnegative.  The output is {x*, v*, y*} where x* is an optimal solution, v* is an optimal value, and y* is a dual optimal solution. It warns if it is unbounded.";
X
XFaces::usage="Faces[list] returns a list of faces for the zero-variable list.  The kth face corresponds to kth inequality in x>=0, m.x<=b, for VertexEnumeration[m,b].";
X
XPolyhedron3D::usage="Polyhedron3D[ver_List,zerova_List] returns 3D graphic primitives for drawing a 3D polytope.  The two arguments correspond to the first two list given by VertexEnumeration[m,b].";
XSubsetQ::usage="SubsetQ[s,t] returns true if s is a subset of t."
X
XAdjacentQ::usage="AdjacentQ[i_Integer,j_Integer,zerova_List] returns True if and only if the zeroset zerova[[i]] and the zeroset zerova[[j]] correspond to adjacent vertices (the intersection of zerova[[i]] and zerova[[j]] is contained in no other zerosets)."
X
XEdgesOfPolyhedron::usage="EdgesOfPolyhedron[zerova_List] returns the set of adjacent pairs of vertices.  See AdjacentQ."
X
XMaximalQ::usage="MaximalQ[i_Integer,l_List] returns True if and only if l[[i]] is a maximal set."
X
XMaximalSets::usage="MaximalSets[l_List] returns the sublist of maximal sets.  See, MaximalQ."
X
XLinesOfArrangement::usage="LinesOfArrangement[zerova_List] returns the list of lines as [1] sets of colinear vertices, and as [2] zerosets."
X
XOptions[VertexEnumeration] = {SearchTree -> False, ZeroVariables -> True, MonitoringFile -> "/dev/null"}
X
XOptions[PointEnumeration] = {SearchTree -> False, ZeroVariables -> True, MonitoringFile -> "/dev/null"}
X
XBegin["`Private`"]
X
XVertexEnumeration::LpInfeasible = "linear program is infeasible."
XVertexEnumeration::LpDualInfeasible = "linear program is dual infeasible."
XVertexEnumeration::LpUnbounded = "linear program is unbounded."
XVertexEnumeration::Infeasible = "linear inequality system is infeasible."
X
X(* comments for private routine *)
XPivot::usage="Pivot[mat,r,s,b,n] performs a pivot operation on mat with row:r and 
Xcolumn:s and updates basis b and nonbasis n. Last 2 arguments are optional.";
X
XGetPivotSequence::usage="GetPivotSequence[startbv,startnbv,goalbv,goalnbv] gives a set of pivots (variable pairs) which transfers the start to the goal.";
X
XSortDict::usage="SortDict[mat,b,n] returns the new {mat,b,n} sorted by b and n reordered in ascending order.";
X
XDic2Vertex::usage="Dic2Vertex[value of bv, name of bv, name of nbv, ver] returns the vertex associated with ver.";
X
XMakeSequence::usage="MakeSequence[dic_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,bases_?MatrixQ] returns pivot sequence.";
X
XGetChangeBaseVariableNames::usage="GetChangeBaseVariableNames[startbv,startnbv,goalbv,goalnbv] gives a set of pivots (variable pairs) which transfers the start to the goal.";
X
XExecPivotSequence::usage="ExecPivotSequence[dic,bv,nbv,sequence] performs a sequence of pivots to {dic,bv,nbv} according to the given sequence."
X
XNonEmptyFaces::usage="NonEmptyFaces[list] returns a list of nonempty faces.";
X
XOptimalBasesEnumeration::usage="OptimalBasesEnumeration[mat,b] gives the set of variable
Xpairs obtaining an optimal basis.";
X
XLexMin::usage="LexMin[mat,bv,nbv,m,n] returns True if the dictionary {mat,bv,nbv} is lexicographically minimum for representing the current solution.";
X
XCrissCrossSelect::usage="CrissCrossSelect[mat,bv,nbv,m,n] gives the pivot position
Xselected by the Criss-Cross rule.";
X
XCrissCrossSelectQ::usage="CrissCrossSelectQ[mat,bv,nbv,i,j] checks whether {i,j} is the pivot position selected by the Criss-Cross rule.";
X
XCrissCrossFilter::usage="CrissCrossFilter[mat,bv,nbv,i,j] is filter (necesarily True) for {i,j}==CrissCrossSelect[mat,bv,nbv].";
X
XCrissCross::usage="CrissCross[mat,bv,nbv,m,n] solves the linear program given by a dictionary {mat,bv,nbv} and outputs a terminal dictionary.";
X
XNegSmallest::usage="Ratio Function used by the Simplex Method";
X
XFirstElement::usage="FirstElement[function,list1[,list2]] gives a position on list1 (ordered by list2),that is True at first time.";
X
XNonNegativeVectorQ::usage="NonNegativeVectorQ[vec] returns True if and only if vec is a nonnegative vector.";
X
XBlandSelect::usage="BlandSelect[mat,bv,nbv] gives a pivot position selected by Bland's rule.";
X
XBlandSelectQ::usage="BlandSelectQ[mat,bv,nbv,i,j] returns True if and only if {i,j}==
XBlandSelect[mat,bv,nbv].";
X
XBlandFilter::usage="BlandFilter[mat,bv,nbv,i,j,m,n] is filter (necesarily True) for {i,j}==
XBlandSelect[mat,bv,nbv].";
X
XBland::usage="Bland[mat,bv,nbv,m,n] solves the linear program given by a feasible dictionary {mat,bv,nbv} and outputs a terminal dictionary.";
X
XDegenerateQ::usage="DegenerateQ[vec] gives True if vec contains 0. It is used to recognize degenerate dictionary.";
X
XInfeasibleQ::usage="InfeasbleQ[mat] gives True if mat is infeasible.";
X
XDualInfeasibleQ::usage="DualInfeasbleQ[mat] gives True if mat is dual infeasible.";
X
XZerovar::usage="Zerovar[Value of bv, Name of bv,Name of nbv] shows the variable names with value 0."
X
XBSearch::usage="BSearch[mat,b,n] returns the set of all basic feasible solutions for the input mat, b, n, {value of bv, name of bv, name of nbv}.";
X
XCSearch::usage="CSearch[mat,b,n] returns the set of all basic solutions for the input mat, b, n, {value of bv, name of bv, name of nbv}.";
X
XOptions[CrissCross] = { NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]]}
X
XOptions[Bland] = { NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]]}
X
XOptions[ZeroVar] = { NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]]}
X
XOptions[BSearch] = {SearchTree -> False, NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]], MonitoringFile->"/dev/null"}
X
XOptions[CSearch] = {SearchTree -> False, NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]], MonitoringFile->"/dev/null"}
X
XVE[mat_?MatrixQ,vec_?VectorQ, opts___Rule]:=VertexEnumeration[mat,vec,opts]
X
XPE[mat_?MatrixQ,vec_?VectorQ, opts___Rule]:=PointEnumeration[mat,vec,opts]
X
XBout2Dlist[bout_List,dlist0_List]:=
X	Block[{i,j,dlist,boutlength,dlist0rule},
X		dlist =  Map[ Function[Father2Edge[Dlist2Father[#[[2]]]]], bout];
X		boutlength = Map[Function[Length[#[[1]]]],bout];
X		boutlength = Prepend[Table[Sum[boutlength[[i]],{i,1,j}],{j,Length[boutlength]-1}],0];
X		dlist += boutlength;
X		dlist0rule = Table[ Rule[i, boutlength[[i]] + 1], {i,Length[boutlength]}];
X		AppendTo[dlist,dlist0 /. dlist0rule];
X		{Flatten[dlist,1],boutlength+1}
X	]
X
X
XVertexEnumeration[mat_?MatrixQ,vec_?VectorQ,opts___Rule]:=
X	Block[{dic,i,j,m,n,bv,nbv,flattenbout,bout,obe,output,dlist,opt1,opt2},
X		opt1 = SearchTree /. {opts} /. Options[VertexEnumeration];
X		opt2 = ZeroVariables /. {opts} /. Options[VertexEnumeration];
X		{m,n} = Dimensions[mat];
X		dic = Transpose[Append[Transpose[-mat],vec]];
X		AppendTo[dic,Append[Table[0,{n}],0]];
X	(* get dic, bv, nbv for Bland. Warnig nbv has been chanded.*)
X		{dic,bv,nbv} = CrissCross[dic,Range[n+1,m+n],Range[n],m+1,n+1,opts];
X	(* feasiblity check *)
X		If[InfeasibleQ[dic], Message[VertexEnumeration::Infeasible];Return[{}]];
X	(* shape up dic for Bland *)
X		dic = Drop[dic,-1];
X		If[DegenerateQ[Last[Transpose[dic]]],
X		(* There is degeneracy *)
X			(* obe looks like {{{dic,bv,nbv},{dic,bv,nbv}}, dlist0} *)
X			obe= OptimalBasesEnumeration[Table[-dic[[i,j]],{i,m},{j,n}],Table[dic[[i,n+1]],{i,m}],bv,nbv];
X			(* bout looks like
X			 {{{{Value of bv,bv,nbv},{Value of bv,bv,nbv},...},dlist},{{{Value of bv,bv,nbv},...},dlist},...} *)
X			bout = Map[ Function[BSearch[#[[1]],#[[2]],#[[3]],opts]],obe[[1]]];
X			(* flattenbout looks like {{Value of bv,bv,nbv},{Value of bv,bv,nbv},....} but no dupilicat startpoint*)
X			flattenbout = Flatten[Map[Function[#[[1]]],bout],1];
X			output = {Map[Function[Dic2Vertex[ #[[1]],#[[2]],#[[3]],Range[n]]],flattenbout]};
X			If[opt2, AppendTo[output, Map[ Function[ZeroVar[#[[1]],#[[2]],#[[3]],opts]], flattenbout]]];
X			If[opt1, 
X				dlist = Bout2Dlist[bout,obe[[2]]];
X				AppendTo[output,dlist[[1]]];
X				AppendTo[output,dlist[[2]]]],
X		(* There is not degeneracy *)
X			(* dic and mat must be same shape *)
X			AppendTo[dic,Append[Table[-1,{n}],0]];
X			bout = BSearch[dic,bv,nbv,opts];
X			output = {Map[ Function[Dic2Vertex[ #[[1]],#[[2]],#[[3]],Range[n]]],bout[[1]]]};
X			If[opt2, AppendTo[output, Map[ Function[ZeroVar[#[[1]],#[[2]],#[[3]],opts]],
X			                            bout[[1]]]]];
X			If[opt1, AppendTo[output, Father2Edge[Dlist2Father[bout[[2]]]]]]
X		];
X		output
X	]
X
X
XPointEnumeration[mat_?MatrixQ,vec_?VectorQ,opts___Rule]:=
X	Block[{dic,i,j,m,n,bv,nbv,tmp,flattenbout,bout,obe,output,opt1,opt2},
X		opt1 = SearchTree /. {opts} /. Options[PointEnumeration];
X		opt2 = ZeroVariables /. {opts} /. Options[PointEnumeration];
X		{m,n} = Dimensions[mat];
X(*
X		dic = Table[
X			Which[
X				SimpleNegative[vec[[i]]], Join[mat[[i]],vec[[i]]],
X				True, Join[-mat[[i]],-vec[[i]]]],
X			{i,m}
X		]
X*)
X		dic = Transpose[Append[Transpose[-mat],vec]];
X		AppendTo[dic,Append[Table[-1,{n}],0]];
X		bv = Range[n+1,m+n];
X		nbv = Range[n];
X		If[DegenerateQ[Last[Transpose[dic]]],
X		(* There is degeneracy *)
X			(* obe looks like {{{dic,bv,nbv},{dic,bv,nbv}}, dlist0} *)
X			obe= OptimalBasesEnumeration[Table[-dic[[i,j]],{i,m},{j,n}],Table[dic[[i,n+1]],{i,m}],bv,nbv];
X			(* bout looks like
X			 {{{{Value of bv,bv,nbv},{Value of bv,bv,nbv},...},dlist},{{{Value of bv,bv,nbv},...},dlist},...} *)
X			bout = Map[ Function[CSearch[#[[1]],#[[2]],#[[3]],opts]],obe[[1]]];
X			(* flattenbout looks like {{Value of bv,bv,nbv},{Value of bv,bv,nbv},....} but no dupilicat startpoint*)
X			flattenbout = Flatten[Map[Function[#[[1]]],bout],1];
X			output = {Map[Function[Dic2Vertex[ #[[1]],#[[2]],#[[3]],Range[n]]],flattenbout]};
X			If[opt2, AppendTo[output, Map[ Function[ZeroVar[#[[1]],#[[2]],#[[3]],opts]], flattenbout]]];
X			If[opt1, 
X				dlist = Bout2Dlist[bout,obe[[2]]];
X				AppendTo[output,dlist[[1]]];
X				AppendTo[output,dlist[[2]]]],
X		(* There is not degeneracy *)
X			(* dic and mat must be same shape *)
X		tmp = CSearch[dic,bv,nbv,opts];
X		output = {Map[ Function[Dic2Vertex[ #[[1]],#[[2]],#[[3]],Range[n]]],tmp[[1]]]};
X		If[opt2, AppendTo[output, Map[ Function[ZeroVar[#[[1]],#[[2]],#[[3]],opts]],
X		                            tmp[[1]]]]];
X		If[opt1, AppendTo[output, Father2Edge[Dlist2Father[tmp[[2]]]]]]
X		];
X		output
X	]
X
XFaces[zerova_List]:=
X	Map[Function[x,
X	      If[Position[zerova,x]=={},{},
X	         Transpose[Position[zerova,x]][[1]]
X	      ]
X	    ], 
X	    Range[Max[zerova]]
X	]
X
XNonEmptyFaces[zerova_List]:=
X	Map[Function[Transpose[Position[zerova,#]][[1]]],Union[Flatten[zerova]]]
X
XMakeLink[ver_List,zerova_List]:=
X		Block[{candidates, cycle ,candidate2cycle},
X		candidates = Drop[ver,1];
X		cycle = { First[ver] };
X		(* when one is list, comparing needs other is list *)
X		While[{} != (candidate2cycle = Select[candidates,AdjacentQ[Last[cycle],#,zerova] &]),
X			AppendTo[cycle,candidate2cycle[[1]]];
X			candidates = Complement[candidates,candidate2cycle[[{1}]]];
X		];
X		While[{} != (candidate2cycle = Select[candidates,AdjacentQ[First[cycle],#,zerova] &]),
X			PrependTo[cycle,candidate2cycle[[1]]];
X			candidates = Complement[candidates,candidate2cycle[[{1}]]];
X		];
X		cycle
X	]
X
XPolyhedron3D[ver_List,zerova_List]:=
X	Polygon /@ (ver[[#]]&) /@ ( MakeLink[#,zerova] & ) /@ NonEmptyFaces[zerova]
X
XZeroVar[ValueOfBv_?VectorQ,bv_?VectorQ,nbv_?VectorQ,opts___Rule]:=
X	Block[{m,out={}},
X		m=Length[ValueOfBv];
X		SimpleNegative := NegativeTest /. {opts} /. Options[ZeroVar];
X		out=Map[Function[SimpleSameQ[#,0]],ValueOfBv];
X		Join[nbv,Map[Function[bv[[#[[1]]]]],Position[out,True]]]
X	]
X		
XForAllQ[func_,list1_List]:= Apply[And, Map[func, list1]]
X	
X
XFather2Edge[fathers_?VectorQ]:=	Map[{#,fathers[[#]]}&,Range[2,Length[fathers]]]
X
XDlist2Father[dlist_?VectorQ]:=
X	Block[{po,dist,father,fathers={1}},
X	(* first element in dist must be 0 and Length[dlist] => 2*)
X		For[po = 2, po <= Length[dlist], ++po,
X			dist =  dlist[[po]];
X			father = po - 1;
X			--dist;
X		(* At first step a element in dist must be 1 *)
X			While[dist > 0,
X				father = fathers[[father]];
X				--dist
X			];
X			AppendTo[fathers,father];
X		];
X		fathers
X	]
X
XDic2Vertex[valueofbv_?VectorQ,bv_?VectorQ,nbv_?VectorQ,ver_?VectorQ]:=
X	Map[Function[If[ MemberQ[nbv,#],
X	                   0,valueofbv[[Position[bv,#][[1,1]]]]]],
X	    ver]
X
X
XLexMin[dic_?MatrixQ,bv_?VectorQ,nbv_?VectorQ]:=
X	Block[{m,n},
X		{m,n}=Dimensions[dic];
X		LexMin[dic,bv,nbv,m,n]
X	]
X
XLexMin[dic_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,m_Integer,n_Integer]:=
X	Block[{i,j},
X		For[i = 1, i < m , ++i,
X			If[SimpleSameQ[0,dic[[i,n]]],
X				For[j = 1, j < n, ++j,
X					If[ Not[SimpleSameQ[0,dic[[i,j]]]] && bv[[i]]  > nbv[[j]], 
X					    Return[False]]
X					]
X				]
X			];
X		True
X	]
X
XCSearch[dict_?MatrixQ,opts___Rule]:=
X	Block[{m,n,bv,nbv},
X		{m,n}=Dimensions[dict];
X		bv = Range[m-1];
X		nbv = Range[m,m+n-2];
X		CSearch[dict,bv,nbv,opts]
X	]
X
XCSearch[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,opts___Rule]:=
X	Block[{ bv=origbv,nbv=orignbv,dict=origdict,m,n,i=1,j=1,opt1,MonitoringFile,
X	        result,dist=1,distlist={0}},
X		opt1 = SearchTree /. {opts} /. Options[CSearch];
X		MonitoringFile = MonitoringFile /. {opts} /. Options[CSearch];
X		SimpleNegative := NegativeTest /. {opts} /. Options[CSearch];
X		{m,n}=Dimensions[dict];
X		If[opt1,
X			result ={{Drop[Transpose[origdict][[n]],-1],origbv,orignbv}};
X			Write[MonitoringFile,Last[result][[3]]],
X			If[LexMin[origdict,origbv,orignbv,m,n], 
X				result ={{Drop[Transpose[origdict][[n]],-1],origbv,orignbv}};
X				Write[MonitoringFile,Last[result][[3]]],
X				result = {}]];
X		While[ ((i < m )|| (origbv != bv)),
X			While[ ((i <  m ) && Not[CrissCrossSelectQ[dict,bv,nbv,i,j]]),
X				++j;
X				If[ j >= n, j = 1; ++i];
X			];
X			If[ i < m,
X				{dict,bv,nbv} = Pivot[dict,bv,nbv,i,j,m,n];
X				If[opt1,
X					AppendTo[result,{Drop[Transpose[dict][[n]],-1],bv,nbv}];
X					Write[MonitoringFile,Last[result][[3]]];
X					AppendTo[distlist,dist];
X					dist = 1,
X					If[LexMin[dict,bv,nbv,m,n],
X						AppendTo[result,{Drop[Transpose[dict][[n]],-1],bv,nbv}];
X						Write[MonitoringFile,Last[result][[3]]];
X						AppendTo[distlist,dist];
X						dist =1;
X					];
X				];
X				{i,j}={1,1},
X				{i,j}=CrissCrossSelect[dict,bv,nbv,m,n];
X				If[ (i < m && j < n),
X					{dict,bv,nbv}=Pivot[dict,bv,nbv,i,j,m,n];
X					++j;
X					++dist;
X					If[ j >= n, j = 1; ++i]
X				];
X			];
X		];
X	{result,distlist}
X	]
X
XBSearch[dict_?MatrixQ,opts___Rule]:=
X	Block[{m,n,bv,nbv},
X		{m,n}=Dimensions[dict];
X		bv = Range[m-1];
X		nbv = Range[m, m+n-2];
X		BSearch[dict,bv,nbv,opts]
X	]
X
XBSearch[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,opts___Rule]:=
X	Block[{ bv=origbv,nbv=orignbv,dict=origdict,m,n,i=1,j=1,opt1,MonitoringFile,
X	        result,dist=1,distlist = {0},bvposi,nbvposi},
X		opt1 = SearchTree /. {opts} /. Options[BSearch];
X		MonitoringFile = MonitoringFile /. {opts} /. Options[BSearch];
X		SimpleNegative := NegativeTest /. {opts} /. Options[BSearch];
X		{m,n}=Dimensions[dict];
X		bvposi = Flatten[Map[Position[bv,#] & ,Sort[bv]]];
X		nbvposi = Flatten[Map[Position[nbv,#] & ,Sort[nbv]]];
X		If[opt1,
X			result ={{Drop[Transpose[origdict][[n]],-1],origbv,orignbv}};
X			Write[MonitoringFile,Last[result][[3]]],
X			If[LexMin[origdict,origbv,orignbv,m,n], 
X				result ={{Drop[Transpose[origdict][[n]],-1],origbv,orignbv}};
X				Write[MonitoringFile,Last[result][[3]]],
X				result = {}]];
X		While[ ((i < m) || (origbv != bv)),
X			While[ (i <  m && Not[BlandSelectQ[dict,bv,nbv,i,j,m,n,bvposi]]),
X				++j;
X				If[ j >= n, j = 1; ++i]
X			];
X			If[ i < m,
X				{dict,bv,nbv} = Pivot[dict,bv,nbv,i,j,m,n];
X				bvposi = Flatten[Map[Position[bv,#] & ,Sort[bv]]];
X				nbvposi = Flatten[Map[Position[nbv,#] & ,Sort[nbv]]];
X				
X(*
X				Print["."];
X*)
X				If[opt1,
X					AppendTo[result,{Drop[Transpose[dict][[n]],-1],bv,nbv}];
X					Write[MonitoringFile,Last[result][[3]]];
X					AppendTo[distlist,dist];
X					dist = 1,
X					If[LexMin[dict,bv,nbv,m,n],
X						AppendTo[result,{Drop[Transpose[dict][[n]],-1],bv,nbv}];
X						Write[MonitoringFile,Last[result][[3]]];
X						AppendTo[distlist,dist];
X						dist = 1,
X					];
X				];
X				{i,j}={1,1},
X				{i,j}=BlandSelect[dict,bv,nbv,bvposi,nbvposi];
X				If[ (i < m && j < n),
X					{dict,bv,nbv}=Pivot[dict,bv,nbv,i,j,m,n];
X					bvposi = Flatten[Map[Position[bv,#] & ,Sort[bv]]];
X					nbvposi = Flatten[Map[Position[nbv,#] & ,Sort[nbv]]];
X					++j;
X					++dist;
X					If[ j >= n, j = 1; ++i]
X				];
X			];
X		];
X	{result,distlist}
X	]
X
XBlandFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer]:=
X	Block[{m,n},
X		{m,n}=Dimensions[dict];
X		BlandFilter[dict,bv,nbv,i,j,m,n,Flatten[Map[Position[bv,#] & ,Sort[bv]]]]
X	]
X
XBlandFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer]:=
X		BlandFilter[dict,bv,nbv,i,j,m,n,Flatten[Map[Position[bv,#] & ,Sort[bv]]]]
X
XBlandFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer,bvposi_?VectorQ]:=
X	( SimpleNegative[dict[[i,j]]] && SimpleNegative[dict[[m,j]]] &&
X	  SimpleSameQ[-dict[[i,n]]/dict[[i,j]],
X	              NegSmallest[Transpose[dict][[j]],bvposi,Last[Transpose[dict]]][[2]]]
X	)
X
XBlandSelectQ[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer]:=
X	BlandSelectQ[origdict,origbv,orignbv,i,j,m,n,Flatten[Map[Position[origbv,#] & ,Sort[origbv]]]]
X	
XBlandSelectQ[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer,bvposi_?VectorQ]:=
X	Block[{dict,bv,nbv},
X		BlandFilter[origdict,origbv,orignbv,i,j,m,n,bvposi] &&
X		(
X		 {dict,bv,nbv}=Pivot[origdict,origbv,orignbv,i,j,m,n];
X		 SameQ[{i,j},BlandSelect[dict,bv,nbv]]
X		)
X	]
X
XNonNegativeVectorQ[vec_?VectorQ]:=
X	ForAllQ[Function[Not[SimpleNegative[#]]],vec]
X
X
XBlandSolve[cost_?VectorQ,mat_?MatrixQ,b_?NonNegativeVectorQ,opts___Rule]:=
X	Block[{dic,m,n,ccdic,ccbv,ccnbv},
X		dic = Transpose[Append[Transpose[Append[-mat,cost]],Append[b,0]]];
X		{m,n}=Dimensions[dic];
X		{ccdic, ccbv, ccnbv}= Bland[dic,opts];
X		If[DualInfeasibleQ[ccdic], Message[VertexEnumeration::LpUnbounded];Return[{}]];
X		{Dic2Vertex[ Drop[Transpose[ccdic][[n]],-1], ccbv, ccnbv,Range[n-1]],ccdic[[m,n]], Dic2Vertex[ Drop[-ccdic[[m]],-1],ccnbv, ccbv, Range[n,n+m-2]]}
X	]
X
XBland[dict_?MatrixQ,opts___Rule]:=
X	Block[{m,n,bv,nbv},
X		{m,n}=Dimensions[dict];
X		bv = Range[n,n+m-2];
X		nbv = Range[n-1];
X		Bland[dict,bv,nbv,m,n,opts]
X	]
X
XBland[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,opts___Rule]:=
X	Block[{m,n},
X		{m,n}=Dimensions[origdict];
X		Bland[origdict,origbv,orignbv,m,n,opts]
X	]
X
XBland[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,m_Integer,n_Integer,opts___Rule]:=
X	Block[{bv=origbv,nbv=orignbv,dict=origdict,i,j},
X		SimpleNegative := NegativeTest /. {opts} /. Options[Bland];
X		For[{i,j}=BlandSelect[dict,bv,nbv], (i < m && j < n),
X		    {i,j}=BlandSelect[dict,bv,nbv],
X		    {dict,bv,nbv}=Pivot[dict,bv,nbv,i,j,m,n]
X		];
X		{dict,bv,nbv}
X	]
X
XBlandSelect[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ]:=
X	Block[{bvposi,nbvposi},
X		bvposi = Flatten[Map[Position[bv,#] & , Sort[bv]]];
X		nbvposi = Flatten[Map[Position[nbv,#] & , Sort[nbv]]];
X		BlandSelect[dict,bv,nbv,bvposi,nbvposi]
X	]
X
XBlandSelect[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,bvposi_?VectorQ,nbvposi_?VectorQ]:=
X	Block[{i,j},
X		j = FirstElement[SimplePositive,Last[dict],nbvposi];
X		i = NegSmallest[
X		                Transpose[dict][[j]],bvposi,
X		                Last[Transpose[dict]]
X		               ][[1]];
X		{i,j}
X	]
X
XCrissCrossFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer]:=
X	Block[{m,n},
X		{m,n}=Dimensions[dict];
X		CrissCrossFilter[dict,bv,nbv,i,j,m,n]
X	]
X
XCrissCrossFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer]:=
X	((
X	 SimplePositive[dict[[i,n]]] && SimplePositive[dict[[i,j]]] && 
X	 ForAllQ[Function[Not[SimpleNegative[dict[i,#]]] &&
X	 nbv[#] < bv[i]],Range[n-1]]
X	)
X	||
X	(
X	 SimpleNegative[dict[[m,j]]] && SimpleNegative[dict[[i,j]]] &&
X	 ForAllQ[Function[Not[SimplePositive[dict[#,j]]] &&
X	 bv[#] < nbv[j]],Range[m-1]]
X	))
X	
XCrissCrossSelectQ[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,i_Integer,j_Integer]:=
X	Block[{dict,bv,nbv,m,n},
X		{m,n}=Dimensions[origdict];
X		CrissCrossFilter[origdict,origbv,orignbv,i,j,m,n] &&
X		(
X		 {dict,bv,nbv}=Pivot[origdict,origbv,orignbv,i,j,m,n];
X		 SameQ[{i,j},CrissCrossSelect[dict,bv,nbv,m,n]]
X		)
X	]
X
XCrissCrossSolve[cost_?VectorQ,mat_?MatrixQ,b_?VectorQ,opts___Rule]:=
X	Block[{dic,m,n,ccdic,ccbv,ccnbv},
X		dic = Transpose[Append[Transpose[Append[-mat,cost]],Append[b,0]]];
X		{m,n}=Dimensions[dic];
X		{ccdic, ccbv, ccnbv}=CrissCross[dic,opts];
X		If[InfeasibleQ[ccdic], Message[VertexEnumeration::LpInfeasible];Return[{}]];
X		If[DualInfeasibleQ[ccdic], Message[VertexEnumeration::LpDualInfeasible];Return[{}]];
X		{Dic2Vertex[ Drop[Transpose[ccdic][[n]],-1], ccbv, ccnbv,Range[n-1]],ccdic[[m,n]], Dic2Vertex[ Drop[-ccdic[[m]],-1],ccnbv, ccbv, Range[n,n+m-2]]}
X	      ]
X
XCrissCross[dict_?MatrixQ,opts___Rule]:=
X	Block[{m,n,bv,nbv},
X		{m,n}=Dimensions[dict];
X		bv = Range[n,m+n-2];
X		nbv = Range[n-1];
X		CrissCross[dict,bv,nbv,m,n,opts]
X	]
X
XCrissCross[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,opts___Rule]:=
X	Block[{m,n},
X		{m,n}=Dimensions[origdict];
X		CrissCross[origdict,origbv,orignbv,m,n,opts]
X	]
X
XCrissCross[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,m_Integer,n_Integer,opts___Rule]:=
X	Block[{bv=origbv,nbv=orignbv,dict=origdict,i,j},
X		SimpleNegative := NegativeTest /. {opts} /. Options[CrissCross];
X		For[{i,j}=CrissCrossSelect[dict,bv,nbv,m,n],(i < m && j < n),
X			{i,j}=CrissCrossSelect[dict,bv,nbv,m,n],
X			{dict,bv,nbv}=Pivot[dict,bv,nbv,i,j,m,n]
X		];
X		{dict,bv,nbv}
X	]
X
XCrissCrossSelect[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ]:=
X	Block[{m,n},
X		{m,n}=Dimensions[dict];
X		CrissCrossSelect[dict,bv,nbv,m,n]
X	]
X
XCrissCrossSelect[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,m_Integer,n_Integer]:=
X	Block[{i,j,bvposi,nbvposi},
X		bvposi = Flatten[Map[ Position[bv,#] & ,Sort[bv]]];
X		nbvposi = Flatten[Map[ Position[nbv,#] & ,Sort[nbv]]];
X		i = FirstElement[SimpleNegative,Last[Transpose[dict]],bvposi];
X		j = FirstElement[SimplePositive,Last[dict],nbvposi];
X		Which[
X			i == m, i = FirstElement[SimpleNegative,Transpose[dict][[j]],bvposi],
X			j == n, j = FirstElement[SimplePositive,dict[[i]],nbvposi],
X			nbv[[j]] > bv[[i]], j = FirstElement[SimplePositive,dict[[i]],nbvposi],
X			True,   i = FirstElement[SimpleNegative,Transpose[dict][[j]],bvposi];
X		];
X		{i,j}
X	]
X	 
XFirstElement[func_,list1_List,list2_List]:=
X	Block[{i},
X		For[i = 1,i <= Length[list2],++i,
X			If[func[list1[[list2[[i]]]]],Return[list2[[i]]]]];
X			i
X	]
X
XFirstElement[func_,list1_List]:=
X	Block[{i},
X		For[i = 1, i <= Length[list1], ++i,
X		If[func[list1[[i]]],Return[i]]];
X		i
X	]
X
XNegSmallest[list1_List,list2_List,list3_List]:=
X	Block[{i,posi,tmpval,val=Infinity},
X		For[i = 1,i <= Length[list2],++i,
X			If[
X				(
X				 SimpleNegative[list1[[list2[[i]]]]] &&
X				 SimpleGreater[val,
X				               tmpval = -(list3[[list2[[i]]]]/list1[[list2[[i]]]])]
X				),
X				posi = i;
X				val = tmpval
X			];
X		];
X		(* val = Infinity -> nothing happens. Never happens *)
X		If[SameQ[Infinity,val], {i,val}, {list2[[posi]],val}]
X	]
X
XPivot[d_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer]:=
X	Block[{tmpbv,tmpnbv},
X		{tmpbv,tmpnbv} = {bv,nbv};
X		{tmpbv[[i]],tmpnbv[[j]]}={tmpnbv[[j]],tmpbv[[i]]};
X		{Pivot[d,i,j,m,n],tmpbv,tmpnbv}
X	]
X
XPivot[d_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer]:=
X	Block[{tmpbv,tmpnbv},
X		{tmpbv,tmpnbv} = {bv,nbv};
X		{tmpbv[[i]],tmpnbv[[j]]}={tmpnbv[[j]],tmpbv[[i]]};
X		{Pivot[d,i,j],tmpbv,tmpnbv}
X	]
X
XPivot[d_?MatrixQ,r_Integer,s_Integer]:=
X	Block[{m,n},
X		{m,n}=Dimensions[d];
X		Pivot[d,r,s,m,n]
X	]
X
XPivot[d_?MatrixQ,r_Integer,s_Integer,m_Integer,n_Integer]:=
X  Block[{i,j},
X	Simplify[
X	  Table[
X	    Which[ (i == r) && (j == s), 1 / d[[i,j]],
X		   (i == r) && (j != s), -(d[[i,j]]/d[[r,s]]),
X		   (i != r) && (j == s), d[[i,j]]/d[[r,s]],
X		   True, d[[i,j]] - ((d[[r,j]] * d[[i,s]]) / d[[r,s]])
X		 ],{i,m},{j,n}
X	       ]
X	     ]
X	   ]
X
X
XGetChangeBaseVariableNames[startbv_?VectorQ,startnbv_?VectorQ,goalbv_?VectorQ,goalnbv_?VectorQ]:=
X	{Intersection[startbv,goalnbv],Intersection[startnbv,goalbv]}
X
XMakeSequence[origdic_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,bases_?MatrixQ]:=
X	Block[{dic = origdic, bv = origbv, nbv = orignbv, bvposi, nbvposi, tmpsequence},
X		(* name to position *)
X		bvposi = Flatten[Map[Position[bv,#] & ,bases[[1]]]];
X		nbvposi = Flatten[Map[Position[nbv,#] & ,bases[[2]]]];
X		(* make all possible sequences *)
X		tmpsequence = Map[Transpose[{bvposi,#}] &, Permutations[nbvposi]];
X		(* choose one *)
X		If[{} == tmpsequence,tmpsequence,tmpsequence[[FirstElement[# &,Map[Apply[And,Map[Not[SimpleSameQ[0,origdic[[#[[1]],#[[2]]]]]] &,#]] &,tmpsequence]]]]]
X	]
X
XGetPivotSequence[startbv_?VectorQ,startnbv_?VectorQ,goalbv_?VectorQ,goalnbv_?VectorQ]:=
X	Transpose[{Intersection[startbv,goalnbv],Intersection[startnbv,goalbv]}]
X
XExecPivotSequence[origdic_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,sequence_?MatrixQ]:=
X	Block[{dic = origdic, bv = origbv, nbv = orignbv, i},
X		Do[{dic,bv,nbv} = Pivot[dic,bv,nbv,sequence[[i,1]],sequence[[i,2]]],{i,Length[sequence]}];
X		{dic,bv,nbv}
X	]
X
XOptimalBasesEnumeration[mat_?MatrixQ,vec_?VectorQ,bv_?VectorQ,nbv_?VectorQ]:=
X	Block[{dic,degenerateBasicVariables,i,m,n,tmp,output},
X		{m,n} = Dimensions[mat];
X		degenerateBasicVariables = Flatten[Position[vec,0]];
X		dic = mat[[degenerateBasicVariables]];
X		AppendTo[dic,Table[1,{n}]];
X		dic = Append[Transpose[dic],Append[Table[-1,{Length[degenerateBasicVariables]}],0]];
X(*
X		bv = Table[i,{i,n+1,m+n}];
X		nbv = Table[i,{i,n}];
X*)
X(* this is dual, so bv and nbv are exchanged *)
X		tmp = BSearch[dic,nbv,bv[[degenerateBasicVariables]],SearchTree->True,MonitoringFile->"/dev/null"];
X		dic = Transpose[Append[Transpose[-mat],vec]];
X		AppendTo[dic,Append[Table[-1,{n}],0]];
X		output = Map[ Function[MakeSequence[dic,bv,nbv,GetChangeBaseVariableNames[bv,nbv,#[[3]],#[[2]]]]], tmp[[1]]];
X		output = {Map[Function[ExecPivotSequence[dic,bv,nbv,#]], output]};
X		AppendTo[output,  Father2Edge[Dlist2Father[tmp[[2]]]]];
X		output
X	]
X
XSubsetQ[s_List,t_List]:=
X   Length[s]==Length[Intersection[s,t]];
X
XAdjacentQ[i_Integer,j_Integer,l_List]:=
X    Block[{subface}, 
X      subface=Intersection[l[[i]],l[[j]]];
X      Length[Select[l,SubsetQ[subface,#]&]]==2]
X
X
XEdgesOfPolyhedron[l_List]:=
X    Block[{i,j,edges={}},
X      Do[
X          Do[If[AdjacentQ[i,j,l],AppendTo[edges,{i,j}]],
X             {j,i+1,Length[l]}
X          ],{i,Length[l]}
X      ];
X    edges]
X
X    
XMaximalQ[i_Integer,l_List]:=
X    Block[{candidate}, 
X      candidate=l[[i]];
X      Length[Union[Select[l,SubsetQ[candidate,#]&]]]==1]
X
XMaximalSets[l_List]:=
X    Block[{i,maximals={}},
X        Do[If[MaximalQ[i,l],
X            AppendTo[maximals,l[[i]]]
X           ], {i,Length[l]}
X        ];
X    maximals]
XLinesOfArrangement[zerova_List]:=
X    Block[{i,j,p=Length[zerova],zeros={},lines={}},
X      Do[Do[AppendTo[zeros,
X              Intersection[zerova[[i]],zerova[[j]]]],{j,i+1,p}
X         ],{i,p}
X      ];zeros=MaximalSets[Union[zeros]];
X      Do[AppendTo[lines,
X           Select[Range[p],SubsetQ[zeros[[i]],zerova[[#]]]&]
X         ],{i,Length[zeros]}
X      ];
X      {lines,zeros} 
X    ]
X
XSortDict[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ]:=
X	Block[{bv=origbv,nbv=orignbv,dict=origdict},
X		dict = Append[
X		              Map[ 
X		                  Function[dict[[Position[bv,#][[1,1]]]]],
X		                  Sort[bv]],
X		              Last[dict]];
X		dict = Append[
X		              Map[
X		                  Function[Transpose[dict][[Position[nbv,#][[1,1]]]]],
X		                  Sort[nbv]],
X		              Last[Transpose[dict]]];
X		{Transpose[dict],Sort[bv],Sort[nbv]}
X	]
X
XDegenerateQ[vec_?VectorQ]:=Count[vec,0]!=0
X
XInfeasibleRowQ[vec_?VectorQ]:=
X	SimpleNegative[Last[vec]] && ForAllQ[Function[Not[SimplePositive[#]]],Drop[vec,-1]]
X
XInfeasibleQ[mat_?MatrixQ]:=
X	Apply[Or,Map[InfeasibleRowQ,Drop[mat,-1]]]
X
XDualInfeasibleQ[mat_?MatrixQ]:=
X	InfeasibleQ[Transpose[-mat]]
X
XSimplePositive:=Function[SimpleNegative[-#]]
X
XSimpleSameQ:=
X	Function[( Not[SimpleNegative[#1 - #2]] && Not[SimplePositive[#1 - #2]] )]
X
XSimpleGreater:=Function[SimplePositive[#1 - #2]]
X
XEnd[(* "`Private`" *)]
X
XProtect[VE]
XProtect[PE]
XProtect[VertexEnumeration]
XProtect[PointEnumeration]
XProtect[CrissCrossSolve]
XProtect[BlandSolve]
XProtect[Faces]
XProtect[Polyhedron3D]
XProtect[SubsetQ]
XProtect[AdjacentQ]
XProtect[EdgesOfPolyhedron]
XProtect[MaximalQ]
XProtect[MaximalSets]
XProtect[LinesOfArrangement]
XEndPackage[]
X
END_OF_FILE
if test 31187 -ne `wc -c <'VertexEnum.m'`; then
    echo shar: \"'VertexEnum.m'\" unpacked with wrong size!
fi
# end of 'VertexEnum.m'
fi
if test -f 'VertexEnum-notebook.ma' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'VertexEnum-notebook.ma'\"
else
echo shar: Extracting \"'VertexEnum-notebook.ma'\" \(258866 characters\)
sed "s/^X//" >'VertexEnum-notebook.ma' <<'END_OF_FILE'
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X	fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L3, e8,  24, "Times"; ;
X	fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6,  18, "Times"; ;
X	fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6,  14, "Times"; ;
X	fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20,  14, "Times"; ;
X	fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15,  12, "Times"; ;
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X	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535,  12, "Courier"; ;
X	fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23,  12, "Courier"; ;
X	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23,  12, "Courier"; ;
X	fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287,  12, "Courier"; ;
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X	fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; ;
X	fontset = Left Header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1,  12, "Times"; ;
X	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7,  12;
X	fontset = Left Footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1,  12, "Times"; ;
X	fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; ;
X	fontset = clipboard, inactive, noKeepOnOnePage, preserveAspect, M7,  12;
X	fontset = completions, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, M7,  12;
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X	fontset = special3, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, right, M7,  12;
X	fontset = special4, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, M7,  12;
X	fontset = special5, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, M7,  12;
X	next21StandardFontEncoding; ]
X:[font = title; inactive; dontPreserveAspect; startGroup; ]
XVertex Enumeration 
Xfor Convex Polyhedra and Hyperplane Arrangements
X- Mathematica Package  -
XVersion 0.41 Beta (November 14, 1992)
X;[s]
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X:[font = subsubtitle; inactive; dontPreserveAspect; ]
XKomei Fukuda and Ichiro Mizukoshi
XGraduate School of Systems Management
XUniversity of Tsukuba, Tokyo
X3-29-1 Otsuka, Bunkyo-ku
XTokyo 112, Japan
X
X+81-3-3942-6876
Xfukuda@gssm.otsuka.tsukuba.ac.jp
Xmizukosi@gssm.otsuka.tsukuba.ac.jp
X
X:[font = text; inactive; dontPreserveAspect; startGroup; ]
XThe package "VertexEnumeration" contains Mathematica implementations of Avis-Fukuda algorithms for enumerating all vertices of a convex polytope given by a system of linear inequalities, and for enumerating all points (0-dimensional faces) of an arrangement of hyperplanes given similarly. 
X;[s]
X3:0,0;41,1;52,2;290,-1;
X3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = text; inactive; dontPreserveAspect; ]
XThis implementation is  quite faithful with the original description of the original algorithms. [1].  
X:[font = text; inactive; dontPreserveAspect; ]
XThe core of this implementation consists of two functions, for a given  s*d  matrix  m  and s-vector  b :  
X
X   (1)  VertexEnumeration[m, b]  
X             -  for computing all vertices of the polyhedron:
X                 P={ x :  x >= 0,  m. x  <=  b } and
X  
X   (2)  PointEnumeration[m,b]
X             -  for computing all points of the arrangements
X                 of  hyperplanes  h_i ,  i= 1,2,...,d+s,   
X                 where   
X                      h_j  = { x : x_j= 0 }  for j=1,....,d,  and
X                      h_(d+i) = { x : m[[i]].x = b_i}  for i=1,...,s.                
X  
XIn particular, these implementions can deal , in principle,  with any rational or floating-point real input with unlimited sizes  s,  d.   Moreover, with some limitations, irrational input handled.  
X;[s]
X23:0,0;85,1;86,2;102,3;103,4;117,5;141,6;222,7;253,8;269,9;291,10;385,11;409,12;460,13;481,14;487,15;497,16;525,17;557,18;563,19;572,20;723,21;728,22;793,-1;
X23:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
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XAs a drawback,  the actual time and space complexity is higher than it was originally stated in [1].  This is partly due to the way Mathematica  is implemented, and also due to our intention to make the program simple.
X;[s]
X3:0,0;132,1;143,2;218,-1;
X3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = text; inactive; dontPreserveAspect; ]
XIncorporating with the standard package Combinatorica   [2] (which is not standard for Version1.2 but available via anonymous FTP from cs.sunysb.edu), one can analyze graph structures associated with convex polytopes and arrangements.
X;[s]
X3:0,0;40,1;53,2;235,-1;
X3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = text; inactive; dontPreserveAspect; endGroup; endGroup; ]
X
X[1] D. Avis and K. Fukuda, 'A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra,'   Proceedings of the 7th ACM Symposium on Computational Geometry, North Conway, New Hampshire, 1991, 98-104.  The full paper is to appear in Discrete and Computational Geometry.
X
X[2] S. S. Skiena, 'Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica,' Addison-Wesley, Reading MA, 1990. 
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X*)
X<<VertexEnum.m;
X(*
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X*)
XOff[General::spell];  Off[General::spell1];   
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XConvex Polyhedra and Linear Inequalities
X:[font = text; inactive; dontPreserveAspect; ]
X3-dimensional convex polyhedra are geometrical objects which are easily understood by drawings.  Mathematica  can draw some well known polyhedra, such as Platonic Solids using the package "Polyhedra.m":
X;[s]
X3:0,0;97,1;108,2;204,-1;
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X:[font = text; inactive; dontPreserveAspect; ]
XHowever,  we cannot draw general convex polyhedra. 
X:[font = text; inactive; dontPreserveAspect; ]
XIn general, a convex polyhedron is defined as the set of solutions to a system of linear inequalities:
X
X(1)                m. x <= b  
X
Xwhere   m   is a real  s*d  matrix and b   is a real s-vector.   3-dimesional polyhera can be represented by such systems (1) with  d=3.   For example, the dodecahedon above is given by a system with s=12 and d=3.
X;[s]
X7:0,0;123,1;134,2;144,3;145,4;175,5;176,6;350,-1;
X7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
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XThe following complex polyhedron is given by a system 
X(1) with a randomly generated matrix  m.
X;[s]
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X:[font = text; inactive; dontPreserveAspect; ]
XIn order to draw such a polyhedron, one must compute all vertices (exremal points) and all edges (adjacencies).  The vertex enumeration package, in fact, computes these for a system of linear inequalities
X
X(2)                m. x <= b   and    x >=0
X
Xwhere a matrix  m  and  a vector b   are  given.   The enumeration of vertices for  a  system  of form (1)  can be easily reduced to that for a system of form (2). 
X;[s]
X7:0,0;225,1;249,2;266,3;268,4;283,5;288,6;416,-1;
X7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = text; inactive; dontPreserveAspect; endGroup; ]
XVery importantly, many fundamental problems can be reduced to the vertex enumeration.   Some of them are the computations of
X    (a) the convex hull of s-points in d-dimensional space,
X    (b) d-dimensional Voronoi diagrams and Delaunay
X          triangulations,
X    (c) optimal solutions to concave programming
X         problems.
XWe shall explain the use of this package through some of these applications.
X 
X:[font = section; inactive; Cclosed; dontPreserveAspect; startGroup; ]
XEnumeration of Vertices
X:[font = subsection; inactive; dontPreserveAspect; startGroup; ]
X3-dimensional Convex Polyhedra
X:[font = text; inactive; dontPreserveAspect; ]
XLet us start with a simple example.    We shall generate a small random matrix  m  of  size * dim.
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Xsize=3;    dim=3;
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Xm=Table[Table[Random[Integer,{1,4}],{dim}],{size}]
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X{{3, 3, 2}, {4, 4, 4}, {1, 4, 2}}
X;[o]
X{{3, 3, 2}, {4, 4, 4}, {1, 4, 2}}
X:[font = text; inactive; dontPreserveAspect; ]
XLet b be the vector of row-sums.
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Xb=Table[Sum[m[[i,j]],{j,dim}],{i,size}]
X
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X{8, 12, 7}
X;[o]
X{8, 12, 7}
X:[font = text; inactive; dontPreserveAspect; ]
XNow we try to identify the shape of the convex polyhedron by using the vertex enumeration package.  (It will take about 1 minute by SE/30.)
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Xvlist=VertexEnumeration[m,b]
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{{{0, 0, 0}, {8/3, 0, 0}, {2, 0, 1}, {0, 0, 3}, 
X   {0, 7/4, 0}, {11/9, 13/9, 0}, {1, 1, 1}, 
X   {0, 1/2, 5/2}}, {{1, 2, 3}, {4, 2, 3}, {4, 2, 5}, 
X   {1, 2, 5}, {1, 6, 3}, {4, 6, 3}, {4, 6, 5}, 
X   {1, 6, 5}}}
X;[o]
X              8
X{{{0, 0, 0}, {-, 0, 0}, {2, 0, 1}, {0, 0, 3}, 
X              3
X 
X       7       11  13                     1  5
X   {0, -, 0}, {--, --, 0}, {1, 1, 1}, {0, -, -}}, 
X       4       9   9                      2  2
X 
X  {{1, 2, 3}, {4, 2, 3}, {4, 2, 5}, {1, 2, 5}, 
X 
X   {1, 6, 3}, {4, 6, 3}, {4, 6, 5}, {1, 6, 5}}}
X:[font = text; inactive; dontPreserveAspect; ]
XThe first part vlist[[1]]   of the output is a list of all vertices, and  the second part vlist[[2]]  is the list of "active-variable" sets, indicating which inequalities are satisfied with equalities at each vertex.   More precisely, for each vertex  v, the active variable set  is  Active(v)={j: h_j(v)=0} , where
X 
X              h_j  = { x : x_j= 0 }  for j=1,....,d,  and
X              h_(d+i) = { x : m[[i]].x = b_i}  for i=1,...,s.
X                
X;[s]
X17:0,0;15,1;25,2;90,3;101,4;259,5;278,6;284,7;307,8;332,9;353,10;359,11;369,12;390,13;421,14;427,15;436,16;455,-1;
X17:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; dontPreserveAspect; ]
Xvertices=vlist[[1]];  activesets=vlist[[2]];
X:[font = text; inactive; dontPreserveAspect; ]
XThe function Polyhedron3D simply generates 3D graphics primitives for drawing the polyhedron. 
X:[font = input; preserveAspect; ]
Xpolytope=Graphics3D[Polyhedron3D[vertices,activesets]];
X:[font = input; dontPreserveAspect; startGroup; ]
XShow[polytope, Boxed->True, ViewPoint->{2,3,1}];
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X:[font = text; inactive; dontPreserveAspect; startGroup; ]
XThe polyderon is determined by several (=size+dim) halfspaces, and bounded by their boundaries (i.e., hyperplanes).  By using PointEnumeration function, we can also compute all points of intersections of hyperplanes.  (The following takes about 2 minutes on SE/30.)
X:[font = input; dontPreserveAspect; startGroup; ]
Xplist=PointEnumeration[m,b]
X:[font = output; output; inactive; dontPreserveAspect; ]
X{{{0, 0, 0}, {8/3, 0, 0}, {0, 0, 4}, {2, 0, 1}, 
X   {3, 0, 0}, {0, 8/3, 0}, {0, 2, 1}, {0, 3, 0}, 
X   {0, 0, 3}, {0, 7/4, 0}, {11/9, 13/9, 0}, 
X   {0, -1, 11/2}, {1, 1, 1}, {5/3, 4/3, 0}, 
X   {1/2, 0, 13/4}, {0, 1/2, 5/2}, {-1, 0, 4}, 
X   {7, 0, 0}, {0, 0, 7/2}}, 
X  {{1, 2, 3}, {4, 2, 3}, {4, 2, 1}, {4, 2, 5}, 
X   {3, 2, 5}, {1, 4, 3}, {1, 4, 5}, {1, 3, 5}, 
X   {1, 2, 5}, {1, 6, 3}, {4, 6, 3}, {4, 6, 1}, 
X   {4, 6, 5}, {3, 6, 5}, {4, 6, 2}, {1, 6, 5}, 
X   {2, 6, 5}, {2, 6, 3}, {1, 6, 2}}}
X;[o]
X              8
X{{{0, 0, 0}, {-, 0, 0}, {0, 0, 4}, {2, 0, 1}, 
X              3
X 
X                  8
X   {3, 0, 0}, {0, -, 0}, {0, 2, 1}, {0, 3, 0}, 
X                  3
X 
X                  7       11  13              11
X   {0, 0, 3}, {0, -, 0}, {--, --, 0}, {0, -1, --}, 
X                  4       9   9               2
X 
X               5  4       1     13       1  5
X   {1, 1, 1}, {-, -, 0}, {-, 0, --}, {0, -, -}, 
X               3  3       2     4        2  2
X 
X                                 7
X   {-1, 0, 4}, {7, 0, 0}, {0, 0, -}}, 
X                                 2
X 
X  {{1, 2, 3}, {4, 2, 3}, {4, 2, 1}, {4, 2, 5}, 
X 
X   {3, 2, 5}, {1, 4, 3}, {1, 4, 5}, {1, 3, 5}, 
X 
X   {1, 2, 5}, {1, 6, 3}, {4, 6, 3}, {4, 6, 1}, 
X 
X   {4, 6, 5}, {3, 6, 5}, {4, 6, 2}, {1, 6, 5}, 
X 
X   {2, 6, 5}, {2, 6, 3}, {1, 6, 2}}}
X:[font = text; inactive; dontPreserveAspect; endGroup; ]
XThen we can draw these points together with the convex polytope to see how the hyperplanes intersect outside the polytope.
X:[font = input; dontPreserveAspect; ]
Xpointplot=Graphics3D[Join[{PointSize[0.01]},
X            Point /@ plist[[1]]]];
X:[font = input; dontPreserveAspect; startGroup; ]
XShow[{pointplot,polytope}, 
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XOne can visualize the partition of the space by the hyperplanes if we draw the lines (1-dimensional flats)  which are intersections of the hyperlanes.
X:[font = input; dontPreserveAspect; startGroup; ]
Xlines=LinesOfArrangement[plist[[2]]];
X:[font = input; dontPreserveAspect; ]
Xplines=(plist[[1]][[#]]&)/@ lines[[1]];
X:[font = input; dontPreserveAspect; ]
Xskeleton=Graphics3D[
X     Join[{Thickness[ 0.002]},
X           Line /@ plines,
X          {PointSize[0.015]},
X           Point /@ plist[[1]]]];
X:[font = input; dontPreserveAspect; startGroup; ]
XShow[{skeleton,polytope},
X     Boxed->False,ViewPoint->{1,3,2}]
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X:[font = output; output; inactive; dontPreserveAspect; endGroup; endGroup; ]
XThe Unformatted text for this cell was not generated.
XUse options in the Actions Preferences dialog box to 
Xcontrol when Unformatted text is generated.
X;[o]
X-Graphics3D-
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X(* 
XDo[Show[{skeleton,polytope},
X   Boxed->False,
X   ViewPoint->{2Cos[theta],2Sin[theta],1}],
X   {theta,0,2Pi,Pi/6}]
X*)
X:[font = subsection; inactive; dontPreserveAspect; startGroup; ]
XGeneral Polyhedra and Graph Structures
X:[font = text; inactive; dontPreserveAspect; ]
XNow we deal with a polyhedron with dimension higher than three.   We need nothing really special for doing this but perhaps more time.  
X
XWe shall generate again a small random matrix  m  of size * dim.
X:[font = input; dontPreserveAspect; startGroup; ]
Xsize=3;    dim=5;
X:[font = input; dontPreserveAspect; startGroup; ]
Xm=Table[Table[Random[Integer,{0,3}],{dim}],{size}]
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{{1, 0, 1, 1, 0}, {0, 0, 2, 0, 1}, {1, 1, 1, 0, 0}}
X;[o]
X{{1, 0, 1, 1, 0}, {0, 0, 2, 0, 1}, {1, 1, 1, 0, 0}}
X:[font = text; inactive; dontPreserveAspect; ]
XLet b  be the vector of all 10's.
X;[s]
X3:0,0;3,1;5,2;34,-1;
X3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; dontPreserveAspect; ]
Xb=Table[10, {size}];
X:[font = text; inactive; dontPreserveAspect; ]
XNow we try to compute the vertices of the convex polyhedron by the vertex enumeration package.  (The following takes roughly 3-5 minutes on Macintosh SE/30 computer.)
X:[font = input; dontPreserveAspect; startGroup; ]
X{time,vlist}=Timing[VertexEnumeration[m,b]]
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{18.75*Second, {{{0, 0, 0, 0, 0}, {10, 0, 0, 0, 0}, 
X    {0, 0, 0, 10, 0}, {0, 0, 5, 0, 0}, 
X    {5, 0, 5, 0, 0}, {10, 0, 0, 0, 10}, 
X    {0, 0, 5, 5, 0}, {0, 0, 0, 10, 10}, 
X    {0, 0, 0, 0, 10}, {0, 10, 0, 10, 10}, 
X    {0, 5, 5, 5, 0}, {0, 5, 5, 0, 0}, 
X    {0, 10, 0, 0, 10}, {0, 10, 0, 10, 0}, 
X    {0, 10, 0, 0, 0}}, 
X   {{1, 2, 3, 4, 5}, {6, 8, 3, 4, 5, 2}, 
X    {1, 2, 3, 6, 5}, {1, 2, 7, 4, 5}, 
X    {6, 8, 7, 4, 5, 2}, {6, 8, 7, 4, 3, 2}, 
X    {1, 2, 7, 6, 5}, {1, 2, 7, 6, 3}, 
X    {1, 2, 7, 4, 3}, {8, 1, 7, 6, 3}, 
X    {8, 1, 7, 6, 5}, {1, 8, 7, 4, 5}, 
X    {1, 8, 7, 4, 3}, {8, 1, 3, 6, 5}, {1, 8, 3, 4, 5}}
X    }}
X;[o]
X{18.75 Second, {{{0, 0, 0, 0, 0}, {10, 0, 0, 0, 0}, 
X 
X    {0, 0, 0, 10, 0}, {0, 0, 5, 0, 0}, 
X 
X    {5, 0, 5, 0, 0}, {10, 0, 0, 0, 10}, 
X 
X    {0, 0, 5, 5, 0}, {0, 0, 0, 10, 10}, 
X 
X    {0, 0, 0, 0, 10}, {0, 10, 0, 10, 10}, 
X 
X    {0, 5, 5, 5, 0}, {0, 5, 5, 0, 0}, 
X 
X    {0, 10, 0, 0, 10}, {0, 10, 0, 10, 0}, 
X 
X    {0, 10, 0, 0, 0}}, 
X 
X   {{1, 2, 3, 4, 5}, {6, 8, 3, 4, 5, 2}, 
X 
X    {1, 2, 3, 6, 5}, {1, 2, 7, 4, 5}, 
X 
X    {6, 8, 7, 4, 5, 2}, {6, 8, 7, 4, 3, 2}, 
X 
X    {1, 2, 7, 6, 5}, {1, 2, 7, 6, 3}, 
X 
X    {1, 2, 7, 4, 3}, {8, 1, 7, 6, 3}, 
X 
X    {8, 1, 7, 6, 5}, {1, 8, 7, 4, 5}, 
X 
X    {1, 8, 7, 4, 3}, {8, 1, 3, 6, 5}, {1, 8, 3, 4, 5}}
X 
X    }}
X:[font = input; dontPreserveAspect; endGroup; ]
Xvertices=vlist[[1]];  activesets=vlist[[2]];
X:[font = text; inactive; dontPreserveAspect; ]
XThe following function computes the edges  of the polyhedron, i.e., the adjacent pairs of vertices.
X;[s]
X3:0,0;35,1;43,2;100,-1;
X3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; dontPreserveAspect; startGroup; ]
Xedges=EdgesOfPolyhedron[activesets]
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{{1, 2}, {1, 3}, {1, 4}, {1, 9}, {1, 15}, {2, 3}, 
X  {2, 5}, {2, 6}, {2, 14}, {2, 15}, {3, 7}, {3, 8}, 
X  {3, 14}, {4, 5}, {4, 7}, {4, 9}, {4, 12}, {5, 6}, 
X  {5, 7}, {5, 11}, {5, 12}, {6, 8}, {6, 9}, {6, 10}, 
X  {6, 13}, {7, 8}, {7, 11}, {8, 9}, {8, 10}, {9, 13}, 
X  {10, 11}, {10, 13}, {10, 14}, {11, 12}, {11, 14}, 
X  {12, 13}, {12, 15}, {13, 15}, {14, 15}}
X;[o]
X{{1, 2}, {1, 3}, {1, 4}, {1, 9}, {1, 15}, {2, 3}, 
X 
X  {2, 5}, {2, 6}, {2, 14}, {2, 15}, {3, 7}, {3, 8}, 
X 
X  {3, 14}, {4, 5}, {4, 7}, {4, 9}, {4, 12}, {5, 6}, 
X 
X  {5, 7}, {5, 11}, {5, 12}, {6, 8}, {6, 9}, {6, 10}, 
X 
X  {6, 13}, {7, 8}, {7, 11}, {8, 9}, {8, 10}, {9, 13}, 
X 
X  {10, 11}, {10, 13}, {10, 14}, {11, 12}, {11, 14}, 
X 
X  {12, 13}, {12, 15}, {13, 15}, {14, 15}}
X:[font = text; inactive; dontPreserveAspect; ]
XAnother important structure we can extract is faces. The Face function gives the list of faces: 
X:[font = input; dontPreserveAspect; startGroup; ]
Xfaces=Faces[activesets]
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{{1, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15}, 
X  {1, 2, 3, 4, 5, 6, 7, 8, 9}, 
X  {1, 2, 3, 6, 8, 9, 10, 13, 14, 15}, 
X  {1, 2, 4, 5, 6, 9, 12, 13, 15}, 
X  {1, 2, 3, 4, 5, 7, 11, 12, 14, 15}, 
X  {2, 3, 5, 6, 7, 8, 10, 11, 14}, 
X  {4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, 
X  {2, 5, 6, 10, 11, 12, 13, 14, 15}}
X;[o]
X{{1, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15}, 
X 
X  {1, 2, 3, 4, 5, 6, 7, 8, 9}, 
X 
X  {1, 2, 3, 6, 8, 9, 10, 13, 14, 15}, 
X 
X  {1, 2, 4, 5, 6, 9, 12, 13, 15}, 
X 
X  {1, 2, 3, 4, 5, 7, 11, 12, 14, 15}, 
X 
X  {2, 3, 5, 6, 7, 8, 10, 11, 14}, 
X 
X  {4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, 
X 
X  {2, 5, 6, 10, 11, 12, 13, 14, 15}}
X:[font = text; inactive; dontPreserveAspect; ]
XThe j-th set faces[[j]]   is the set of vertices lying on the hyperplaneh_j,   where
X 
X         h_j  = { x : x_j= 0 }  for j=1,....,dim,  and
X         h_(dim+i) = { x : m[[i]].x = b_i}  for i=1,...,size.
X
XEach one is a face but may not be a facet.  The facets are the maximal faces.
X;[s]
X13:0,0;13,1;23,2;72,3;79,4;96,5;117,6;123,7;135,8;151,9;184,10;190,11;202,12;283,-1;
X13:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; dontPreserveAspect; startGroup; ]
Xfacets=MaximalSets[faces]
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{{1, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15}, 
X  {1, 2, 3, 4, 5, 6, 7, 8, 9}, 
X  {1, 2, 3, 6, 8, 9, 10, 13, 14, 15}, 
X  {1, 2, 4, 5, 6, 9, 12, 13, 15}, 
X  {1, 2, 3, 4, 5, 7, 11, 12, 14, 15}, 
X  {2, 3, 5, 6, 7, 8, 10, 11, 14}, 
X  {4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, 
X  {2, 5, 6, 10, 11, 12, 13, 14, 15}}
X;[o]
X{{1, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15}, 
X 
X  {1, 2, 3, 4, 5, 6, 7, 8, 9}, 
X 
X  {1, 2, 3, 6, 8, 9, 10, 13, 14, 15}, 
X 
X  {1, 2, 4, 5, 6, 9, 12, 13, 15}, 
X 
X  {1, 2, 3, 4, 5, 7, 11, 12, 14, 15}, 
X 
X  {2, 3, 5, 6, 7, 8, 10, 11, 14}, 
X 
X  {4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, 
X 
X  {2, 5, 6, 10, 11, 12, 13, 14, 15}}
X:[font = text; inactive; dontPreserveAspect; ]
XIf one wants to compute the adjacency among the facets,  simply use the function  EdgesOfPolyhedron,  again.
X:[font = input; dontPreserveAspect; startGroup; ]
Xcoedges=EdgesOfPolyhedron[facets]
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, 
X  {1, 8}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, 
X  {3, 4}, {3, 5}, {3, 6}, {3, 7}, {3, 8}, {4, 5}, 
X  {4, 7}, {4, 8}, {5, 6}, {5, 7}, {5, 8}, {6, 7}, 
X  {6, 8}, {7, 8}}
X;[o]
X{{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, 
X 
X  {1, 8}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, 
X 
X  {3, 4}, {3, 5}, {3, 6}, {3, 7}, {3, 8}, {4, 5}, 
X 
X  {4, 7}, {4, 8}, {5, 6}, {5, 7}, {5, 8}, {6, 7}, 
X 
X  {6, 8}, {7, 8}}
X:[font = text; inactive; dontPreserveAspect; startGroup; ]
XWith help of the combinatorica package,  you can draw and analyze the graph structure of the polyhedron. 
X:[font = text; inactive; dontPreserveAspect; ]
X(* The next drawing requires  Combinatorica.m  package. *) 
X:[font = input; dontPreserveAspect; startGroup; ]
X<<Combinatorica.m;    (* Optional *)
X:[font = message; inactive; preserveAspect; endGroup; ]
XGet::noopen: Can't open Combinatorica.m.
X:[font = input; dontPreserveAspect; startGroup; ]
XShowLabeledGraph[gpoly=FromUnorderedPairs[edges]]
X:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 17; pictureWidth = 282; pictureHeight = 282; ]
X%!
X%%Creator: Mathematica
X%%AspectRatio: 1 
XMathPictureStart
X/Courier findfont 10  scalefont  setfont
X% Scaling calculations
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X[ 1 1 0 0 ]
X] MathScale
X% Start of Graphics
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Xnewpath
X%%Object: Graphics
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XMathPictureEnd
X:[font = text; inactive; dontPreserveAspect; ]
XLet us make a small example by generating few random points in the plane.
X:[font = input; dontPreserveAspect; ]
Xdim=2;  size=5;  min=1;  max=30;  width=max-min;
X:[font = input; dontPreserveAspect; startGroup; ]
Xdatapoints=
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X  
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{{10, 18}, {20, 23}, {13, 17}, {16, 7}, {26, 14}}
X;[o]
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XLet us draw these points using 3D graphics.  (2D graphics is sufficient for this purpose but 3D graphics will be useful later.)
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XIn order to compute the Voronoi diagram for the given points, we compute the vertices of 3-dimensional convex polyhedron and then project its boundary to a plane.   The convex polyhedron corresponds to a piecewise-linear approximation of a parabola:  z=x^2+y^2.  Each piece is associated with some datapoint  {x_i, y_i}  and is a part of the hyperplane supporting the parabola at this point.   This hyperplane is represented by  z=2*x + 2*y + (x_i^2 + y_i^2).  The convex polyhedron is then represented by the linear inequalities:
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X:[font = input; dontPreserveAspect; ]
X(*  The above is generated by 
Xparabola=Plot3D[x^2+y^2,{x,0,max},{y,0,max},
X    BoxRatios->{1,1,1},  ViewPoint->{1,-2,1}]    
X*)   
X:[font = text; inactive; dontPreserveAspect; ]
X
XWe make the input matrix  m  and vector  b  for the vertex enumeration package for the convex polyhedron.  Here notice that we will compute this within the square 
X[{x, min, max}, {y, min, max}].   Also, the parabola will be lifted so that the approximation won't touch the bottom plane z=0.
X:[font = input; dontPreserveAspect; startGroup; ]
Xm1=Map[Append[2 #,-1]&,datapoints];
X;[s]
X1:0,0;35,-1;
X1:1,10,8,Times,1,12,0,0,0;
X:[font = input; dontPreserveAspect; ]
Xb0=Map[Sum[#[[j]]^2,{j,dim}]&,datapoints];
X;[s]
X1:0,0;42,-1;
X1:1,10,8,Times,1,12,0,0,0;
X:[font = input; dontPreserveAspect; ]
Xblift=2*Min[b0];   b1=Map[(#-blift)&,b0];
X;[s]
X1:0,0;41,-1;
X1:1,10,8,Times,1,12,0,0,0;
X:[font = input; dontPreserveAspect; ]
Xm=Join[m1,{{-1,0,0},{0,-1,0},{1,0,0},{0,1,0}}];
X;[s]
X1:0,0;47,-1;
X1:1,10,8,Times,1,12,0,0,0;
X:[font = input; dontPreserveAspect; endGroup; ]
Xb=Join[b1,{-min,-min,max,max}];
X;[s]
X1:0,0;31,-1;
X1:1,10,8,Times,1,12,0,0,0;
X:[font = text; inactive; dontPreserveAspect; ]
X
XPlease notice that the main computation below takes 3-4 minutes on SE/30.
X:[font = input; dontPreserveAspect; startGroup; ]
X{time,vlist}=Timing[VertexEnumeration[m,b]]
X;[s]
X1:0,0;43,-1;
X1:1,10,8,Times,1,12,0,0,0;
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{18.98333333333333*Second, 
X  {{{1, 131/22, 4624/11}, {493/54, 187/18, 20050/27}, 
X    {4599/242, 3231/242, 133106/121}, 
X    {433/22, 1075/66, 40178/33}, {27/2, 47/2, 1302}, 
X    {41/4, 30, 1471}, {1, 30, 1286}, {30, 30, 2261}, 
X    {1, 1, 351}, {553/20, 1, 6019/5}, {30, 1, 1326}, 
X    {30, 139/6, 5840/3}}, 
X   {{9, 7, 4}, {6, 7, 4}, {6, 7, 8}, {6, 5, 8}, 
X    {6, 5, 4}, {12, 5, 4}, {12, 9, 4}, {12, 5, 11}, 
X    {9, 7, 10}, {8, 7, 10}, {8, 11, 10}, {8, 11, 5}}}}
X;[o]
X                       131  4624    493  187  20050
X{18.9833 Second, {{{1, ---, ----}, {---, ---, -----}, 
X                       22    11     54   18    27
X 
X     4599  3231  133106    433  1075  40178
X    {----, ----, ------}, {---, ----, -----}, 
X     242   242    121      22    66    33
X 
X     27  47          41
X    {--, --, 1302}, {--, 30, 1471}, {1, 30, 1286}, 
X     2   2           4
X 
X                                  553     6019
X    {30, 30, 2261}, {1, 1, 351}, {---, 1, ----}, 
X                                  20       5
X 
X                        139  5840
X    {30, 1, 1326}, {30, ---, ----}}, 
X                         6    3
X 
X   {{9, 7, 4}, {6, 7, 4}, {6, 7, 8}, {6, 5, 8}, 
X 
X    {6, 5, 4}, {12, 5, 4}, {12, 9, 4}, {12, 5, 11}, 
X 
X    {9, 7, 10}, {8, 7, 10}, {8, 11, 10}, {8, 11, 5}}}}
X:[font = text; inactive; dontPreserveAspect; ]
XWe can see the voronoi diagram by projecting the polyhedral surface onto x-y plane.   Of course, this can be simulated by looking the surface from the sky.
X:[font = input; dontPreserveAspect; ]
Xpoly=Graphics3D[
X       Drop[Polyhedron3D[vlist[[1]],vlist[[2]]],-4]];
X:[font = input; dontPreserveAspect; startGroup; ]
XShow[{poly,dataplot},
X     ViewPoint->{0,0,3}, Shading->True,
X     Boxed->False, BoxRatios->{1,1,0}];
X:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 17; pictureWidth = 282; pictureHeight = 282; endGroup; ]
X%!
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X:[font = text; inactive; dontPreserveAspect; ]
XIn order to compute the graph structure of the Voronoi diagram, one must do a little more.
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XVoronoiAdjacency=EdgesOfPolyhedron[vlist[[2]]];
X;[s]
X1:0,0;47,-1;
X1:1,10,8,Times,1,12,0,0,0;
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X;[s]
X1:0,0;187,-1;
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XWe can now draw the Voronoi diagram, the datapoints and the polyhedral surface at the same time.
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X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
XThe Unformatted text for this cell was not generated.
XUse options in the Actions Preferences dialog box to 
Xcontrol when Unformatted text is generated.
X;[o]
X-Graphics3D-
X:[font = text; inactive; dontPreserveAspect; ]
XThe Delaunay triangulation comes hand-in-hand with the Voronoi diagam.  What we have to do is to compute the adjacency among the Voronoi regions (facets of the associated polyhedron.)
X:[font = input; dontPreserveAspect; startGroup; ]
Xfacets=Faces[vlist[[2]]];
X;[s]
X1:0,0;25,-1;
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X;[s]
X1:0,0;34,-1;
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XDelaunayAdjacency=Select[coedges,
X    (dim+1<#[[1]]<=size+dim+1 && 
X     dim+1<#[[2]]<=size+dim+1)&] 
X;[s]
X1:0,0;101,-1;
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X{{4, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 8}, {6, 7}, {6, 8}, 
X  {7, 8}}
X;[o]
X{{4, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 8}, {6, 7}, {6, 8}, 
X 
X  {7, 8}}
X:[font = input; dontPreserveAspect; ]
XDelaunayEdges=(points[[#-dim-1]]&) /@ DelaunayAdjacency; 
X;[s]
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X         {PointSize[0.02],GrayLevel[0]}, 
X         Point /@ points]];
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XShow[{VoronoiDiagram,Delaunay},
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XGeneral Diagrams
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XGeneral (d-dimensional) Voronoi diagrams (tasselations) can be computed in the exactly same manner as for the plane, through computing the structure of a (d+1)-dimensional convex polyhedron.
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X{{8, 9, 17, 8}, {14, 10, 12, 14}, {15, 12, 7, 9}, 
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X;[o]
X{{8, 9, 17, 8}, {14, 10, 12, 14}, {15, 12, 7, 9}, 
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Xm=m1;  b=b1;
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XWe can compute the vertices of  the polyhedron whose boundary represents the Voronoi partitions.   Notice that the computation will be restricted to the nonnegative orthant and thus the result may not contain the  complete structure of the partition.
X:[font = text; inactive; dontPreserveAspect; ]
XThe next one takes 3-4 minutes on  SE/30.
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XDelaunayEdges=(#-dim-1)& /@ DelaunayAdjacency;
X:[font = text; inactive; dontPreserveAspect; ]
XWe can draw the Delaunay structure (the adjacency among the Voronoi regions) by using Combinatorica package.
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X(*  << Combinatorica.m;    *)
X
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XConvex Hull
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X3-dimensional Convex Hull
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XComputing the convex hull of a given set of points can be indirectly done by the vertex enumeration.   In this section, we see how this can be done through a 3-dimensional example.
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X;[o]
X{{0, 1, -1}, {-2, 2, -2}, {-2, -2, -2}, {-1, 2, 1}, 
X 
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X9
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XThe convex hull of given points is the smallest convex region (convex polyhedron) containing the points.   
X
XThe theory relating the convex hull and the vertex enumeration is the polarity.   Let  P  be a convex polyhedron containing the origin in its interior.   Let  P*  be the convex set defined by  {y :  x.y  <= 1  for all  x  in  P}.  Then P* is also a convex polyhedron whose face lattice is anti-isomorphic to that of  P.   Therefore, there is a one-to-one correspondance between the vertices of  P*  and the facets of  P  preserving the adjacency.
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XThus, what we must do first is to translate the given points so that the origin will be contained in the convex hull.
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X******  Caution:   Long computation  ******
XThe next one takes 4-5 minutes on  SE/30.
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XWe can easily compute which datapoints form a facet of the convex hull.  
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X:[font = subsection; inactive; Cclosed; dontPreserveAspect; startGroup; ]
XGeneral Convex Hull
X:[font = text; inactive; dontPreserveAspect; endGroup; endGroup; ]
XHere, we just note that the convex hull of points in general d-space can be computed through the same operation (polarization) and the vertex enumeration.
X   
X:[font = section; inactive; Cclosed; dontPreserveAspect; startGroup; ]
XOptions for Enumeration
X:[font = subsection; inactive; dontPreserveAspect; startGroup; ]
XMonitoring the Computation
X:[font = text; inactive; dontPreserveAspect; ]
XSometimes, you may want to monitor how the computation is being done.   For this purpose, one can use  MonitoringFile  option, which allows you to output the list of nonbasic varibles as the computation proceeds.   The output can be directed to a file you specify, or to CRT if the filename is "stdout".
X;[s]
X5:0,0;102,1;117,2;295,3;301,4;304,-1;
X5:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; dontPreserveAspect; ]
Xm={{1,1,1},{0,1,0},{0,0,1}};
X:[font = input; dontPreserveAspect; ]
Xb={1,1,1};
X:[font = input; dontPreserveAspect; startGroup; ]
Xvlist=VertexEnumeration[m,b,MonitoringFile->"stdout"]
X:[font = print; inactive; dontPreserveAspect; ]
X{1, 2, 3}
X{4, 2, 3}
X{4, 5, 3}
X{4, 2, 6}
X:[font = output; output; inactive; dontPreserveAspect; endGroup; endGroup; ]
X{{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, 
X  {{1, 2, 3}, {4, 2, 3}, {4, 5, 3, 1}, {4, 2, 6, 1}}}
X;[o]
X{{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, 
X 
X  {{1, 2, 3}, {4, 2, 3}, {4, 5, 3, 1}, {4, 2, 6, 1}}}
X:[font = subsection; inactive; Cclosed; dontPreserveAspect; startGroup; ]
XOutput All Basic Solutions
X:[font = text; inactive; dontPreserveAspect; ]
XInstead of outputing all vertices without repetition, one can list all basic solutions with possible repetitions of vertices.   For this purpose one can use the SearchTree option.   If this option is set  True (default is False), VertexEnumeration outputs all basic feasible solutions together with the Search Tree structure which the Avis-Fukuda algorithm actually traces.
X;[s]
X5:0,0;160,1;171,2;205,3;210,4;374,-1;
X5:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; dontPreserveAspect; ]
Xm={{1,1,1},{0,1,0},{0,0,1}};
X:[font = input; dontPreserveAspect; ]
Xb={1,1,1};
X:[font = input; dontPreserveAspect; startGroup; ]
Xvlist=VertexEnumeration[m,b,SearchTree->True,
X             MonitoringFile->"stdout"]
X:[font = print; inactive; dontPreserveAspect; ]
X{1, 2, 3}
X{4, 2, 3}
X{1, 4, 3}
X{1, 2, 4}
X{1, 5, 3}
X{4, 5, 3}
X{1, 5, 4}
X{1, 2, 6}
X{4, 2, 6}
X{1, 4, 6}
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X{{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {0, 1, 0}, 
X   {0, 1, 0}, {0, 1, 0}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1}}, 
X  {{1, 2, 3}, {4, 2, 3}, {1, 4, 3, 5}, {1, 2, 4, 6}, 
X   {1, 5, 3, 4}, {4, 5, 3, 1}, {1, 5, 4, 3}, {1, 2, 6, 4}, 
X   {4, 2, 6, 1}, {1, 4, 6, 2}}, 
X  {{2, 1}, {3, 1}, {4, 1}, {5, 1}, {6, 5}, {7, 5}, {8, 1}, 
X   {9, 8}, {10, 8}}}
X;[o]
X{{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {0, 1, 0}, 
X 
X   {0, 1, 0}, {0, 1, 0}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1}}, 
X 
X  {{1, 2, 3}, {4, 2, 3}, {1, 4, 3, 5}, {1, 2, 4, 6}, 
X 
X   {1, 5, 3, 4}, {4, 5, 3, 1}, {1, 5, 4, 3}, {1, 2, 6, 4}, 
X 
X   {4, 2, 6, 1}, {1, 4, 6, 2}}, 
X 
X  {{2, 1}, {3, 1}, {4, 1}, {5, 1}, {6, 5}, {7, 5}, {8, 1}, 
X 
X   {9, 8}, {10, 8}}}
X:[font = text; inactive; dontPreserveAspect; ]
XHere is a way to draw the Avis-Fukuda search tree  by using a nice embedding function of Combinatorica.
X:[font = input; dontPreserveAspect; ]
XSTree=vlist[[3]];
X:[font = input; dontPreserveAspect; startGroup; ]
XShowLabeledGraph[RankedEmbedding[FromUnorderedPairs[STree],{1}]]
X:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 17; pictureWidth = 282; pictureHeight = 282; ]
X%!
X%%Creator: Mathematica
X%%AspectRatio: 1 
XMathPictureStart
X% Scaling calculations
X-1.1 2 0.05556 1.11111 [
X[ 0 0 0 0 ]
X[ 1 1 0 0 ]
X] MathScale
X% Start of Graphics
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X%%Object: Graphics
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X[(10)] 0.84 0.8 0 1 Mshowa
Xgrestore
X% End of Graphics
XMathPictureEnd
X:[font = output; output; inactive; dontPreserveAspect; endGroup; endGroup; endGroup; ]
XThe Unformatted text for this cell was not generated.
XUse options in the Actions Settings dialog box to control
Xwhen Unformatted text is generated.
X;[o]
X-Graphics-
X:[font = section; inactive; Cclosed; dontPreserveAspect; startGroup; ]
XDealing with Irrational Data in Mathematica 1.2
X;[s]
X3:0,0;32,1;43,2;47,-1;
X3:1,12,9,Times,1,14,0,0,0;1,13,10,Times,3,14,0,0,0;1,12,9,Times,1,14,0,0,0;
X:[font = text; inactive; dontPreserveAspect; ]
XMathematica  can handle irrational numbers, but we must be careful because some older versions (Version 1.2 or earlier) do not simplify some trivial expressions like 
X;[s]
X2:0,0;11,1;167,-1;
X2:1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; dontPreserveAspect; startGroup; ]
Xir = 5* Sqrt[5] - 5^(3/2)  
X:[font = output; output; inactive; dontPreserveAspect; endGroup; ]
X5*5^(1/2) - 5^(3/2)
X;[o]
X   1/2    3/2
X5 5    - 5
X:[font = text; inactive; dontPreserveAspect; ]
Xwhich is obviously zero.   This may cause some serious problems if you want to use irrational numbers in input data.    The separate small program "Power.m" can be used with VertexEnumeration and PointEnumeration to prevent such situations with irrational numbers of rational power type  p^q   where both p and q are rational numbers.
X:[font = input; dontPreserveAspect; ]
Xm={{ir +1, ir + 5^(1/3), 1}};   b={1};
X:[font = input; dontPreserveAspect; ]
X<<Power.m;    (* this is for Mathematica Version 1.2 *)
X;[s]
X3:0,0;29,1;40,2;56,-1;
X3:1,10,8,Courier,1,12,0,0,0;1,10,8,Courier,3,12,0,0,0;1,10,8,Courier,1,12,0,0,0;
X:[font = input; dontPreserveAspect; startGroup; ]
Xvlist=VertexEnumeration[m,b]
X:[font = output; output; inactive; dontPreserveAspect; endGroup; endGroup; ]
X{{{0, 0, 0}, {0, 5^(-1/3), 0}, {1, 0, 0}, {0, 0, 1}}, 
X  {{1, 2, 3}, {1, 4, 3}, {2, 4, 3}, {1, 4, 2}}}
X;[o]
X                  -(1/3)
X{{{0, 0, 0}, {0, 5      , 0}, {1, 0, 0}, {0, 0, 1}}, 
X 
X  {{1, 2, 3}, {1, 4, 3}, {2, 4, 3}, {1, 4, 2}}}
X:[font = section; inactive; Cclosed; dontPreserveAspect; startGroup; ]
XSupplementary Functions
X:[font = text; inactive; preserveAspect; ]
XThere are some useful functions defined in the VertexEnum package.   We shall introduce some of those below.
X:[font = subsection; inactive; dontPreserveAspect; startGroup; ]
XLinear Programming
X:[font = text; inactive; preserveAspect; ]
XThe problem of maximizing a linear function over a convex polyhedron is known to be Linear programming.   A standard form of linear programming problem (LP) is  maximize  c.x  subject to m.x <=b and x>=0, where  m  is a given  s x d matrix,  c and b are  given d-vector and s-vector, respectively.    The function CrissCrossSolve solves this LP.
X;[s]
X3:0,0;161,1;203,2;346,-1;
X3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; preserveAspect; startGroup; ]
X?CrissCrossSolve
X:[font = info; inactive; preserveAspect; endGroup; ]
XCrissCrossSolve[c,m,b] solves the linear program 
X   maximize c.x subject to m.x<=b and x>=0 by the
X   Criss-Cross method. The output is {x*, v*, y*} where
X   x* is an optimal solution, v* is an optimal value, and
X   y* is a dual optimal solution. It warns if it is
X   infeasible or dualinfeasible
X:[font = input; preserveAspect; startGroup; ]
Xm={{4, -3, 3}, {-2, 1, -3}};
X:[font = input; preserveAspect; endGroup; ]
Xc={1,-2,-4};  b={5,-3};
X:[font = input; preserveAspect; startGroup; ]
Xoptimum=CrissCrossSolve[c,m,b]
X:[font = output; output; inactive; preserveAspect; endGroup; ]
X{{2, 1, 0}, 0, {3/2, 5/2}}
X;[o]
X                3  5
X{{2, 1, 0}, 0, {-, -}}
X                2  2
X:[font = text; inactive; preserveAspect; ]
XCrissCrossSolve always finds a vertex solution if the LP has an optimal solution.  You can verify this by using VetexEnumeration. 
X:[font = input; preserveAspect; startGroup; ]
XVertexEnumeration[m,b]
X:[font = output; output; inactive; preserveAspect; endGroup; ]
X{{{2, 1, 0}, {1, 0, 1/3}, {0, 0, 5/3}, {0, 0, 1}}, 
X  {{5, 4, 3}, {5, 4, 2}, {1, 4, 2}, {5, 1, 2}}}
X;[o]
X                    1          5
X{{{2, 1, 0}, {1, 0, -}, {0, 0, -}, {0, 0, 1}}, 
X                    3          3
X 
X  {{5, 4, 3}, {5, 4, 2}, {1, 4, 2}, {5, 1, 2}}}
X:[font = text; inactive; preserveAspect; ]
XMathematica has also LinearProgramming function which solves LP.  One has to be little careful because some earliear versions of LinearProgramming function (Version 2.0 or earlier) do not work correctly.   Also this function solves a different form of LP, which is dual to our formulation.
X;[s]
X2:0,0;11,1;290,-1;
X2:1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; preserveAspect; startGroup; ]
X?LinearProgramming
X:[font = info; inactive; preserveAspect; endGroup; ]
XLinearProgramming[c, m, b] finds the vector x which
X   minimizes the quantity c.x subject to the constraints
X   m.x >= b and x >= 0.
X:[font = text; inactive; preserveAspect; ]
XIn order to solve the same LP as above, we must use the negatives of the input arguments.
X:[font = input; preserveAspect; startGroup; ]
XLinearProgramming[-c,-m,-b]
X:[font = output; output; inactive; preserveAspect; endGroup; ]
X{2, 1, 0}
X;[o]
X{2, 1, 0}
X:[font = text; inactive; preserveAspect; ]
XWhile LinearProgramming function often runs faster for rational inputs, it does not allow any irrational inputs.   It is an advantage of CrissCrossSolve that irrational inputs are allowed.  (The next one takes about 1 minute on SE/30.)
X:[font = input; preserveAspect; startGroup; ]
Xm={{1, Pi, E}, {E, 1, Pi}}; 
Xb={Log[5],Sqrt[3]};  c={E,Pi,Sqrt[5]};
Xopt=CrissCrossSolve[c,m,b]
X:[font = output; output; inactive; preserveAspect; endGroup; ]
X{{(3^(1/2)*Pi - Log[5])/(-1 + E*Pi), 
X   (-3^(1/2) + E*Log[5])/(-1 + E*Pi), 0}, 
X  (-(3^(1/2)*Pi) + 3^(1/2)*E*Pi - E*Log[5] + 
X     E*Pi*Log[5])/(-1 + E*Pi), 
X  {-((E*(-1 + Pi))/(1 - E*Pi)), 
X   -(((1 - E)*Pi)/(-1 + E*Pi))}}
X;[o]
X  Sqrt[3] Pi - Log[5]  -Sqrt[3] + E Log[5]
X{{-------------------, -------------------, 0}, 
X       -1 + E Pi            -1 + E Pi
X 
X  -(Sqrt[3] Pi) + Sqrt[3] E Pi - E Log[5] + E Pi Log[5]
X  -----------------------------------------------------, 
X                        -1 + E Pi
X 
X     E (-1 + Pi)     (1 - E) Pi
X  {-(-----------), -(----------)}}
X      1 - E Pi       -1 + E Pi
X:[font = text; inactive; preserveAspect; ]
XFor LPs with no optimal solutions, CrissCrossSolve outputs the following warnings.
X:[font = input; preserveAspect; startGroup; ]
XCrissCrossSolve[{1,1},{{2,3},{-2,-3}},{2,-4}]
X:[font = message; inactive; preserveAspect; ]
XVertexEnumeration::LpInfeasible: 
X   linear program is infeasible.
X:[font = output; output; inactive; preserveAspect; endGroup; ]
X{}
X;[o]
X{}
X:[font = input; preserveAspect; startGroup; ]
XCrissCrossSolve[{1,1},{{-2,3}},{-4}]
X:[font = message; inactive; preserveAspect; ]
XVertexEnumeration::LpDualInfeasible: 
X   linear program is dual infeasible.
X:[font = output; output; inactive; preserveAspect; endGroup; ]
X{}
X;[o]
X{}
X:[font = text; inactive; preserveAspect; endGroup; endGroup; ]
XIt should be noted that the dual infeasibility implies that the original problem is either unbounded or infeasible, i.e., cannot have an optimal solution.
X:[font = section; inactive; preserveAspect; startGroup; ]
XList of Exported Functions and Options
X:[font = input; preserveAspect; startGroup; ]
X?VertexEnum`*
X:[font = info; inactive; preserveAspect; endGroup; endGroup; ]
XAdjacentQ          MaximalQ           SearchTree
XBlandSolve         MaximalSets        SubsetQ
XCrissCrossSolve    MonitoringFile     VE
XEdgesOfPolyhedron  PE                 VertexEnumeration
XFaces              PointEnumeration   ZeroVariables
XLinesOfArrangement Polyhedron3D
X:[font = text; inactive; preserveAspect; ]
X(* This is the end of VertexEnum-notebook  *)
X
END_OF_FILE
if test 258866 -ne `wc -c <'VertexEnum-notebook.ma'`; then
    echo shar: \"'VertexEnum-notebook.ma'\" unpacked with wrong size!
fi
# end of 'VertexEnum-notebook.ma'
fi
if test -f 'Power.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'Power.m'\"
else
echo shar: Extracting \"'Power.m'\" \(155 characters\)
sed "s/^X//" >'Power.m' <<'END_OF_FILE'
XUnprotect[Power];
XPower[x_?NumberQ,y_]:= ( x ) * Power[x, y - 1] /; y > 1 ; 
XPower[x_?NumberQ,y_]:= ( 1 / x) * Power[x, y + 1] /; y < -1;
XProtect[Power];
X
END_OF_FILE
if test 155 -ne `wc -c <'Power.m'`; then
    echo shar: \"'Power.m'\" unpacked with wrong size!
fi
# end of 'Power.m'
fi
if test -f 'FaceLattice.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'FaceLattice.m'\"
else
echo shar: Extracting \"'FaceLattice.m'\" \(3946 characters\)
sed "s/^X//" >'FaceLattice.m' <<'END_OF_FILE'
X(*
X  FaceLattice.m: Face Enumeration for Convex Polytopes
X     
X             Version 0.2 Beta
X              September 19, 1992
X
X           Copyright (c) 1992 by
X
X  Komei Fukuda
X  Graduate School of Systems Management
X  University of Tsukuba, Tokyo
X  3-29-1 Otsuka, Bunkyo-ku
X  Tokyo, Japan 112
X  +81-3-3942-6876
X  fukuda@gssm.otsuka.tsukuba.ac.jp
X
X  and
X  
X  Vera Rosta
X  Department of Mathematics
X  Temple University Japan
X  2-2 Minami-Osawa, Hachioji-shi
X  Tokyo, Japan 192-03
X
X*)
X
XBeginPackage["FaceLattice`", "VertexEnum`", "DiscreteMath`Combinatorica`"]
X
XKFaceList::usage = "KFaceList[vlist] gives the list of k-faces of
Xthe convex polytope P with combinatorial representation of vertices 
Xgiven by  vlist, for k=-1,0,1,...,dim(P)."
X
XFVector::usage = "FVector[kfl] gives the f-vector of 
Xthe convex polytope with k-face list kfl."
X
XFaceLatticeDiagram::usage = "FaceLatticeDiagram[kfl,psize,ewidth] gives 
Xthe graphics primitive of the face lattice of 
Xthe convex polytope with k-face list kfl.  
XPoint size and edge width are both optional."
X
XFaceLatticeLinks::usage = "FaceLatticeLinks[kfl] gives 
Xthe links (covering relation) of the face lattice of 
Xthe convex polytope with k-face list kfl."
X
XBegin["`Private`"]
X
XKFaceList[l_List]:=
X    Block[{allfaces={{Apply[Union,l]}},
X           ktemp=Map[Union[#]&,l],kfaces,k1faces={},
X           doflag,ftemp,s,i,j,k=0},
X      While[Length[ktemp]>1,
X      	 doflag=Table[True,{Length[ktemp]}];
X         Do[
X         	If[doflag[[i]],
X         		Do[
X         			If[doflag[[j]],
X         				ftemp=Union[Intersection[ktemp[[i]],ktemp[[j]]]];
X         				If[k1faces=={} || Position[k1faces,ftemp]=={},
X         					k1faces=Append[k1faces,ftemp];
X         					s=Position[ktemp,ftemp];
X             					If[s!={},
X         						doflag[[s[[1,1]]]]=False
X         					]
X         				]
X         			]
X         			,{j,i+1,Length[ktemp]}
X         		]
X         	]
X            ,{i,Length[ktemp]-1}
X         ];
X         kfaces={};
X         Do[
X         	If[doflag[[i]],
X         		AppendTo[kfaces,ktemp[[i]]]
X         	]
X         	,{i,Length[ktemp]}
X         ];
X         Print["All ",k,"-faces determined."];
X         allfaces=Append[allfaces,kfaces];
X         ktemp=k1faces;   k1faces={};  k=k+1
X      ];
X      allfaces=Append[allfaces,ktemp]
X    ];
X
XFVector[kfl_List]:=Map[Length[#]&,kfl];
X
XFaceLatticeLinks[kfl_List]:=
X    Block[{k,d,i,j,f1,f2,allrel={},krel},
X      d=Length[kfl]-2;
X      Do[krel={};
X         Do[
X            Do[f1=kfl[[k+1,i]];  f2=kfl[[k+2,j]];
X               If[Complement[f2,f1]=={},
X                  krel=Join[krel,
X                    {{{k-1,i},{k,j}}}],
X               ],
X               {j,Length[kfl[[k+2]]]}
X            ],
X            {i,Length[kfl[[k+1]]]}
X         ];
X         allrel=Join[allrel,krel],  
X         {k,0,d}
X      ];
X      allrel
X    ];
X
XFacePosition[kfl_List,k_Integer,j_Integer]:=
X   Block[{x,y,fk,fmax,fmin,width,d=(Length[kfl]-2)},
X      fmax=Max[FVector[kfl]];
X      fmin=Min[FVector[kfl]];
X      fk=Length[kfl[[k+2]]];
X      If[fmax>fmin,
X         width=2/3+(1/3)(fk-fmin)/(fmax-fmin),1];
X      If[fk<=1,x=1/2,x=(1-width)/2+(j-1)*width/(fk-1)]; 
X      If[d<=-1,0,y=(k+1)/(d+1)];
X      {x,y}
X   ];
X
X
XFaceLatticeEdges[kfl_List,flink_List]:=
X   Map[{FacePosition[kfl,#[[1,1]],#[[1,2]]],
X        FacePosition[kfl,#[[2,1]],#[[2,2]]]}&,
X      flink
X   ];
X    
XFaceLatticePoints[kfl_List]:=
X  Flatten[
X     Table[
X        Table[FacePosition[kfl,k-2,j],
X           {j,Length[kfl[[k]]]}
X        ],
X        {k,Length[kfl]}
X     ],1
X  ];
X  
X
XFaceLatticeDiagram[kfl_List,psize_:0.01,ewidth_:0.002]:=
X	Block[{HasseLinks,EdgePositions,PointPositions},
X		HasseLinks=FaceLatticeLinks[kfl];
X		EdgePositions=FaceLatticeEdges[kfl,HasseLinks];
X		PointPositions=FaceLatticePoints[kfl];
X		Graphics[
X 			Join[{Thickness[ewidth],GrayLevel[0]},
X 		   		Line /@ EdgePositions,
X				{PointSize[psize],GrayLevel[0]},
X				Point /@ PointPositions
X			]
X		]
X	]
X
X  
XEnd[]
X
XEndPackage[]
X
END_OF_FILE
if test 3946 -ne `wc -c <'FaceLattice.m'`; then
    echo shar: \"'FaceLattice.m'\" unpacked with wrong size!
fi
# end of 'FaceLattice.m'
fi
if test -f 'FaceLattice-notebook.ma' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'FaceLattice-notebook.ma'\"
else
echo shar: Extracting \"'FaceLattice-notebook.ma'\" \(48676 characters\)
sed "s/^X//" >'FaceLattice-notebook.ma' <<'END_OF_FILE'
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X:[font = title; inactive; preserveAspect; ]
XGenerating All Faces of Convex Polytopes:FaceLattice.m
X-Mathematica  Package -
XVersion 0.2 Beta,  September 19, 1992
X
XKomei Fukuda(University of Tsukuba, Tokyo) 
Xand 
XVera Rosta(Temple University Japan)
X;[s]
X5:0,0;56,1;67,2;116,3;117,4;203,-1;
X5:1,21,16,Times,1,24,0,0,0;1,13,10,Times,3,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,10,8,Courier,1,12,0,0,0;1,12,9,Times,1,14,0,0,0;
X:[font = text; inactive; preserveAspect; ]
XIn this small notebook, we shall explain how to enumerate all faces of a d-dimensional convex polytope using the package FaceLattice.m which is distributed with VertexEnum package. (A convex polytope is a bounded convex polyhedron.)   Since the package FaceLattice.m  automatically reads two other packages, VertexEnum.m and Combinatorica.m, it is important that those packages are available in your computing environment before you try read in the FaceLattice package.  Combinatorica.m is a standard Mathematica package which is to be in the directory DiscreteMath. 
X;[s]
X5:0,0;205,1;212,2;501,3;512,4;568,-1;
X5:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; initialization; preserveAspect; ]
X*)
X<<FaceLattice.m;
X(*
X:[font = input; initialization; preserveAspect; ]
X*)
XOff[General::spell];  Off[General::spell1];   
X(*
X:[font = text; inactive; preserveAspect; startGroup; ]
XLet us first generate randomly a system of linear inequalities: m.x <=b and x>=0 whose solution set  is a 5-dimensional convex polytope with 7 facets.
X:[font = input; dontPreserveAspect; startGroup; ]
Xsize=2;    dim=5;
X:[font = input; dontPreserveAspect; startGroup; ]
Xm=Table[Table[Random[Integer,{1,5}],{dim}],{size}]
X:[font = output; input; inactive; dontPreserveAspect; endGroup; ]
X{{5, 1, 2, 3, 4}, {2, 5, 3, 2, 5}}
X;[o]
X{{5, 1, 2, 3, 4}, {2, 5, 3, 2, 5}}
X:[font = input; dontPreserveAspect; startGroup; ]
Xb=Table[Sum[m[[i,j]],{j,dim}],{i,size}]
X:[font = output; input; inactive; dontPreserveAspect; endGroup; endGroup; ]
X{15, 17}
X;[o]
X{15, 17}
X:[font = text; inactive; preserveAspect; ]
XThen we compute all vertices of the polytope using the VetexEnumeration function of  VertexEnum package. 
X:[font = input; dontPreserveAspect; ]
Xvlist=VertexEnumeration[m,b];
X:[font = input; dontPreserveAspect; startGroup; ]
Xactivesets=vlist[[2]]
X:[font = output; output; inactive; preserveAspect; endGroup; ]
X{{1, 2, 3, 4, 5}, {6, 2, 3, 4, 5}, {6, 2, 3, 1, 5}, 
X  {1, 7, 3, 4, 5}, {1, 7, 2, 4, 5}, {1, 2, 3, 4, 7}, 
X  {6, 2, 3, 4, 7}, {6, 2, 3, 1, 7}, {6, 5, 3, 4, 7}, 
X  {6, 5, 3, 1, 7}, {6, 5, 2, 4, 7}, {6, 5, 2, 1, 7}}
X;[o]
X{{1, 2, 3, 4, 5}, {6, 2, 3, 4, 5}, {6, 2, 3, 1, 5}, 
X 
X  {1, 7, 3, 4, 5}, {1, 7, 2, 4, 5}, {1, 2, 3, 4, 7}, 
X 
X  {6, 2, 3, 4, 7}, {6, 2, 3, 1, 7}, {6, 5, 3, 4, 7}, 
X 
X  {6, 5, 3, 1, 7}, {6, 5, 2, 4, 7}, {6, 5, 2, 1, 7}}
X:[font = text; inactive; preserveAspect; ]
XHere activesets is the list of vertices, each represented as the set of indices of inequalities which are satisfied by equality (i.e. active) at the vertex. We can represent each face of the polytope exactly the same manner.  In fact, by taking all possible intersections of members of activesets, we can generate all faces. The function KFaceList outputs {F-1,F0,F1,F2,...,Fd} where d is the dimension of the polytope and Fk is the list of k-dimensional faces for k=-1,0,1,2,...,d.
X;[s]
X7:0,0;5,1;15,2;134,3;140,4;338,5;347,6;483,-1;
X7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = text; inactive; preserveAspect; ]
X(* The following takes about two minutes on Macintosh SE/30 *)
X:[font = input; Cclosed; preserveAspect; startGroup; ]
X{time,flist}=Timing[KFaceList[activesets]]
X:[font = print; inactive; preserveAspect; ]
XAll 0-faces determined.
XAll 1-faces determined.
XAll 2-faces determined.
XAll 3-faces determined.
XAll 4-faces determined.
X:[font = output; output; inactive; preserveAspect; endGroup; ]
X{10.15*Second, {{{1, 2, 3, 4, 5, 6, 7}}, 
X   {{1, 2, 3, 4, 5}, {2, 3, 4, 5, 6}, {1, 2, 3, 5, 6}, 
X    {1, 3, 4, 5, 7}, {1, 2, 4, 5, 7}, {1, 2, 3, 4, 7}, 
X    {2, 3, 4, 6, 7}, {1, 2, 3, 6, 7}, {3, 4, 5, 6, 7}, 
X    {1, 3, 5, 6, 7}, {2, 4, 5, 6, 7}, {1, 2, 5, 6, 7}}, 
X   {{2, 3, 4, 5}, {1, 2, 3, 5}, {1, 3, 4, 5}, 
X    {1, 2, 4, 5}, {1, 2, 3, 4}, {2, 3, 5, 6}, 
X    {2, 3, 4, 6}, {3, 4, 5, 6}, {2, 4, 5, 6}, 
X    {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 2, 5, 6}, 
X    {1, 4, 5, 7}, {1, 3, 4, 7}, {3, 4, 5, 7}, 
X    {1, 3, 5, 7}, {1, 2, 4, 7}, {2, 4, 5, 7}, 
X    {1, 2, 5, 7}, {2, 3, 4, 7}, {1, 2, 3, 7}, 
X    {2, 3, 6, 7}, {3, 4, 6, 7}, {2, 4, 6, 7}, 
X    {1, 3, 6, 7}, {1, 2, 6, 7}, {3, 5, 6, 7}, 
X    {4, 5, 6, 7}, {1, 5, 6, 7}, {2, 5, 6, 7}}, 
X   {{2, 3, 5}, {3, 4, 5}, {2, 4, 5}, {2, 3, 4}, 
X    {1, 3, 5}, {1, 2, 5}, {1, 2, 3}, {1, 4, 5}, 
X    {1, 3, 4}, {1, 2, 4}, {2, 3, 6}, {3, 5, 6}, 
X    {2, 5, 6}, {3, 4, 6}, {2, 4, 6}, {4, 5, 6}, 
X    {1, 3, 6}, {1, 2, 6}, {1, 5, 6}, {1, 4, 7}, 
X    {4, 5, 7}, {1, 5, 7}, {3, 4, 7}, {1, 3, 7}, 
X    {3, 5, 7}, {2, 4, 7}, {1, 2, 7}, {2, 5, 7}, 
X    {2, 3, 7}, {3, 6, 7}, {2, 6, 7}, {4, 6, 7}, 
X    {1, 6, 7}, {5, 6, 7}}, 
X   {{3, 5}, {2, 5}, {2, 3}, {4, 5}, {3, 4}, {2, 4}, 
X    {1, 5}, {1, 3}, {1, 2}, {1, 4}, {3, 6}, {2, 6}, 
X    {5, 6}, {4, 6}, {1, 6}, {4, 7}, {1, 7}, {5, 7}, 
X    {3, 7}, {2, 7}, {6, 7}}, 
X   {{5}, {3}, {2}, {4}, {1}, {6}, {7}}, {{}}}}
X;[o]
X{10.15 Second, {{{1, 2, 3, 4, 5, 6, 7}}, 
X 
X   {{1, 2, 3, 4, 5}, {2, 3, 4, 5, 6}, {1, 2, 3, 5, 6}, 
X 
X    {1, 3, 4, 5, 7}, {1, 2, 4, 5, 7}, {1, 2, 3, 4, 7}, 
X 
X    {2, 3, 4, 6, 7}, {1, 2, 3, 6, 7}, {3, 4, 5, 6, 7}, 
X 
X    {1, 3, 5, 6, 7}, {2, 4, 5, 6, 7}, {1, 2, 5, 6, 7}}, 
X 
X   {{2, 3, 4, 5}, {1, 2, 3, 5}, {1, 3, 4, 5}, 
X 
X    {1, 2, 4, 5}, {1, 2, 3, 4}, {2, 3, 5, 6}, 
X 
X    {2, 3, 4, 6}, {3, 4, 5, 6}, {2, 4, 5, 6}, 
X 
X    {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 2, 5, 6}, 
X 
X    {1, 4, 5, 7}, {1, 3, 4, 7}, {3, 4, 5, 7}, 
X 
X    {1, 3, 5, 7}, {1, 2, 4, 7}, {2, 4, 5, 7}, 
X 
X    {1, 2, 5, 7}, {2, 3, 4, 7}, {1, 2, 3, 7}, 
X 
X    {2, 3, 6, 7}, {3, 4, 6, 7}, {2, 4, 6, 7}, 
X 
X    {1, 3, 6, 7}, {1, 2, 6, 7}, {3, 5, 6, 7}, 
X 
X    {4, 5, 6, 7}, {1, 5, 6, 7}, {2, 5, 6, 7}}, 
X 
X   {{2, 3, 5}, {3, 4, 5}, {2, 4, 5}, {2, 3, 4}, 
X 
X    {1, 3, 5}, {1, 2, 5}, {1, 2, 3}, {1, 4, 5}, 
X 
X    {1, 3, 4}, {1, 2, 4}, {2, 3, 6}, {3, 5, 6}, 
X 
X    {2, 5, 6}, {3, 4, 6}, {2, 4, 6}, {4, 5, 6}, 
X 
X    {1, 3, 6}, {1, 2, 6}, {1, 5, 6}, {1, 4, 7}, 
X 
X    {4, 5, 7}, {1, 5, 7}, {3, 4, 7}, {1, 3, 7}, 
X 
X    {3, 5, 7}, {2, 4, 7}, {1, 2, 7}, {2, 5, 7}, 
X 
X    {2, 3, 7}, {3, 6, 7}, {2, 6, 7}, {4, 6, 7}, 
X 
X    {1, 6, 7}, {5, 6, 7}}, 
X 
X   {{3, 5}, {2, 5}, {2, 3}, {4, 5}, {3, 4}, {2, 4}, 
X 
X    {1, 5}, {1, 3}, {1, 2}, {1, 4}, {3, 6}, {2, 6}, 
X 
X    {5, 6}, {4, 6}, {1, 6}, {4, 7}, {1, 7}, {5, 7}, 
X 
X    {3, 7}, {2, 7}, {6, 7}}, 
X 
X   {{5}, {3}, {2}, {4}, {1}, {6}, {7}}, {{}}}}
X:[font = text; inactive; preserveAspect; ]
XThe functin FVector counts the number of k-faces for k=-1,0,2,...,d.
X:[font = input; preserveAspect; startGroup; ]
Xfvector=FVector[flist]
X:[font = output; input; inactive; preserveAspect; endGroup; ]
X{1, 12, 30, 34, 21, 7, 1}
X;[o]
X{1, 12, 30, 34, 21, 7, 1}
X:[font = text; inactive; preserveAspect; ]
XIt is still open question whether or not the f-vector of any convex polytope is unimodal, i.e., there is only one peak in the sequece.  On the other hand there is a well-known theorem about the f-vector:  Euler-Poincare's relation.  This relation says that the alternate sum of the number of k-faces is always zero.  We can test whether it is true for our example.  First we make the (+1,-1) alternating vector.  
X:[font = input; preserveAspect; startGroup; ]
Xavector=Table[(-1)^i,{i,Length[flist]}]
X:[font = output; output; inactive; preserveAspect; endGroup; ]
X{-1, 1, -1, 1, -1, 1, -1}
X;[o]
X{-1, 1, -1, 1, -1, 1, -1}
X:[font = text; inactive; preserveAspect; ]
XThe following must be true.  (If not, who is responsible for it?)
X:[font = input; preserveAspect; startGroup; ]
XTrueQ[avector.fvector==0]
X:[font = output; output; inactive; preserveAspect; endGroup; ]
XTrue
X;[o]
XTrue
X:[font = text; inactive; preserveAspect; ]
XIt is not easy to draw by hand the face lattice of general convex polytopes, but the function FaceLatticeDiagram automatically generates a Mathematica  graphics for it.  Whether or not the drawing is useful is a different question!
X;[s]
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X:[font = input; preserveAspect; startGroup; ]
XShow[FaceLatticeDiagram[flist]]
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X%!
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X:[font = output; output; inactive; preserveAspect; endGroup; ]
XThe Unformatted text for this cell was not generated.
XUse options in the Actions Preferences dialog box to 
Xcontrol when Unformatted text is generated.
X;[o]
X-Graphics-
X:[font = text; inactive; preserveAspect; ]
XIf  you want to know exactly what are the links in the diagram above, execute the function  FaceLatticeLinks.
X;[s]
X3:0,0;92,1;108,2;110,-1;
X3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;
X:[font = input; preserveAspect; startGroup; ]
Xflinks=FaceLatticeLinks[flist];
Xflinks //Short
X:[font = output; output; inactive; preserveAspect; endGroup; ]
XShort[{{{-1, 1}, {0, 1}}, {{-1, 1}, {0, 2}}, 
X   {{-1, 1}, {0, 3}}, {{-1, 1}, {0, 4}}, 
X   {{-1, 1}, {0, 5}}, {{-1, 1}, {0, 6}}, 
X   {{-1, 1}, {0, 7}}, {{-1, 1}, {0, 8}}, 
X   {{-1, 1}, {0, 9}}, {{-1, 1}, {0, 10}}, 
X   {{-1, 1}, {0, 11}}, {{-1, 1}, {0, 12}}, 
X   {{0, 1}, {1, 1}}, {{0, 1}, {1, 2}}, 
X   {{0, 1}, {1, 3}}, {{0, 1}, {1, 4}}, 
X   {{0, 1}, {1, 5}}, {{0, 2}, {1, 1}}, 
X   {{0, 2}, {1, 6}}, {{0, 2}, {1, 7}}, 
X   {{0, 2}, {1, 8}}, {{0, 2}, {1, 9}}, 
X   {{0, 3}, {1, 2}}, {{0, 3}, {1, 6}}, 
X   {{0, 3}, {1, 10}}, {{0, 3}, {1, 11}}, 
X   {{0, 3}, {1, 12}}, {{0, 4}, {1, 3}}, 
X   {{0, 4}, {1, 13}}, {{0, 4}, {1, 14}}, 
X   {{0, 4}, {1, 15}}, {{0, 4}, {1, 16}}, 
X   {{0, 5}, {1, 4}}, {{0, 5}, {1, 13}}, 
X   {{0, 5}, {1, 17}}, {{0, 5}, {1, 18}}, 
X   {{0, 5}, {1, 19}}, {{0, 6}, {1, 5}}, 
X   {{0, 6}, {1, 14}}, {{0, 6}, {1, 17}}, 
X   {{0, 6}, {1, 20}}, {{0, 6}, {1, 21}}, 
X   {{0, 7}, {1, 7}}, {{0, 7}, {1, 20}}, 
X   {{0, 7}, {1, 22}}, {{0, 7}, {1, 23}}, 
X   {{0, 7}, {1, 24}}, {{0, 8}, {1, 10}}, 
X   {{0, 8}, {1, 21}}, {{0, 8}, {1, 22}}, 
X   {{0, 8}, {1, 25}}, {{0, 8}, {1, 26}}, 
X   {{0, 9}, {1, 8}}, {{0, 9}, {1, 15}}, 
X   {{0, 9}, {1, 23}}, {{0, 9}, {1, 27}}, 
X   {{0, 9}, {1, 28}}, {{0, 10}, {1, 11}}, 
X   {{0, 10}, {1, 16}}, {{0, 10}, {1, 25}}, 
X   {{0, 10}, {1, 27}}, {{0, 10}, {1, 29}}, 
X   {{0, 11}, {1, 9}}, {{0, 11}, {1, 18}}, 
X   {{0, 11}, {1, 24}}, {{0, 11}, {1, 28}}, 
X   {{0, 11}, {1, 30}}, {{0, 12}, {1, 12}}, 
X   {{0, 12}, {1, 19}}, {{0, 12}, {1, 26}}, 
X   {{0, 12}, {1, 29}}, {{0, 12}, {1, 30}}, 
X   {{1, 1}, {2, 1}}, {{1, 1}, {2, 2}}, 
X   {{1, 1}, {2, 3}}, {{1, 1}, {2, 4}}, 
X   {{1, 2}, {2, 1}}, {{1, 2}, {2, 5}}, 
X   {{1, 2}, {2, 6}}, {{1, 2}, {2, 7}}, 
X   {{1, 3}, {2, 2}}, {{1, 3}, {2, 5}}, 
X   {{1, 3}, {2, 8}}, {{1, 3}, {2, 9}}, 
X   {{1, 4}, {2, 3}}, {{1, 4}, {2, 6}}, 
X   {{1, 4}, {2, 8}}, {{1, 4}, {2, 10}}, 
X   {{1, 5}, {2, 4}}, {{1, 5}, {2, 7}}, 
X   {{1, 5}, {2, 9}}, {{1, 5}, {2, 10}}, 
X   {{1, 6}, {2, 1}}, {{1, 6}, {2, 11}}, 
X   {{1, 6}, {2, 12}}, {{1, 6}, {2, 13}}, 
X   {{1, 7}, {2, 4}}, {{1, 7}, {2, 11}}, 
X   {{1, 7}, {2, 14}}, {{1, 7}, {2, 15}}, 
X   {{1, 8}, {2, 2}}, {{1, 8}, {2, 12}}, 
X   {{1, 8}, {2, 14}}, {{1, 8}, {2, 16}}, 
X   {{1, 9}, {2, 3}}, {{1, 9}, {2, 13}}, 
X   {{1, 9}, {2, 15}}, {{1, 9}, {2, 16}}, 
X   {{1, 10}, {2, 7}}, {{1, 10}, {2, 11}}, 
X   {{1, 10}, {2, 17}}, {{1, 10}, {2, 18}}, 
X   {{1, 11}, {2, 5}}, {{1, 11}, {2, 12}}, 
X   {{1, 11}, {2, 17}}, {{1, 11}, {2, 19}}, 
X   {{1, 12}, {2, 6}}, {{1, 12}, {2, 13}}, 
X   {{1, 12}, {2, 18}}, {{1, 12}, {2, 19}}, 
X   {{1, 13}, {2, 8}}, {{1, 13}, {2, 20}}, 
X   {{1, 13}, {2, 21}}, {{1, 13}, {2, 22}}, 
X   {{1, 14}, {2, 9}}, {{1, 14}, {2, 20}}, 
X   {{1, 14}, {2, 23}}, {{1, 14}, {2, 24}}, 
X   {{1, 15}, {2, 2}}, {{1, 15}, {2, 21}}, 
X   {{1, 15}, {2, 23}}, {{1, 15}, {2, 25}}, 
X   {{1, 16}, {2, 5}}, {{1, 16}, {2, 22}}, 
X   {{1, 16}, {2, 24}}, {{1, 16}, {2, 25}}, 
X   {{1, 17}, {2, 10}}, {{1, 17}, {2, 20}}, 
X   {{1, 17}, {2, 26}}, {{1, 17}, {2, 27}}, 
X   {{1, 18}, {2, 3}}, {{1, 18}, {2, 21}}, 
X   {{1, 18}, {2, 26}}, {{1, 18}, {2, 28}}, 
X   {{1, 19}, {2, 6}}, {{1, 19}, {2, 22}}, 
X   {{1, 19}, {2, 27}}, {{1, 19}, {2, 28}}, 
X   {{1, 20}, {2, 4}}, {{1, 20}, {2, 23}}, 
X   {{1, 20}, {2, 26}}, {{1, 20}, {2, 29}}, 
X   {{1, 21}, {2, 7}}, {{1, 21}, {2, 24}}, 
X   {{1, 21}, {2, 27}}, {{1, 21}, {2, 29}}, 
X   {{1, 22}, {2, 11}}, {{1, 22}, {2, 29}}, 
X   {{1, 22}, {2, 30}}, {{1, 22}, {2, 31}}, 
X   {{1, 23}, {2, 14}}, {{1, 23}, {2, 23}}, 
X   {{1, 23}, {2, 30}}, {{1, 23}, {2, 32}}, 
X   {{1, 24}, {2, 15}}, {{1, 24}, {2, 26}}, 
X   {{1, 24}, {2, 31}}, {{1, 24}, {2, 32}}, 
X   {{1, 25}, {2, 17}}, {{1, 25}, {2, 24}}, 
X   {{1, 25}, {2, 30}}, {{1, 25}, {2, 33}}, 
X   {{1, 26}, {2, 18}}, {{1, 26}, {2, 27}}, 
X   {{1, 26}, {2, 31}}, {{1, 26}, {2, 33}}, 
X   {{1, 27}, {2, 12}}, {{1, 27}, {2, 25}}, 
X   {{1, 27}, {2, 30}}, {{1, 27}, {2, 34}}, 
X   {{1, 28}, {2, 16}}, {{1, 28}, {2, 21}}, 
X   {{1, 28}, {2, 32}}, {{1, 28}, {2, 34}}, 
X   {{1, 29}, {2, 19}}, {{1, 29}, {2, 22}}, 
X   {{1, 29}, {2, 33}}, {{1, 29}, {2, 34}}, 
X   {{1, 30}, {2, 13}}, {{1, 30}, {2, 28}}, 
X   {{1, 30}, {2, 31}}, {{1, 30}, {2, 34}}, 
X   {{2, 1}, {3, 1}}, {{2, 1}, {3, 2}}, 
X   {{2, 1}, {3, 3}}, {{2, 2}, {3, 1}}, 
X   {{2, 2}, {3, 4}}, {{2, 2}, {3, 5}}, 
X   {{2, 3}, {3, 2}}, {{2, 3}, {3, 4}}, 
X   {{2, 3}, {3, 6}}, {{2, 4}, {3, 3}}, 
X   {{2, 4}, {3, 5}}, {{2, 4}, {3, 6}}, 
X   {{2, 5}, {3, 1}}, {{2, 5}, {3, 7}}, 
X   {{2, 5}, {3, 8}}, {{2, 6}, {3, 2}}, 
X   {{2, 6}, {3, 7}}, {{2, 6}, {3, 9}}, 
X   {{2, 7}, {3, 3}}, {{2, 7}, {3, 8}}, 
X   {{2, 7}, {3, 9}}, {{2, 8}, {3, 4}}, 
X   {{2, 8}, {3, 7}}, {{2, 8}, {3, 10}}, 
X   {{2, 9}, {3, 5}}, {{2, 9}, {3, 8}}, 
X   {{2, 9}, {3, 10}}, {{2, 10}, {3, 6}}, 
X   {{2, 10}, {3, 9}}, {{2, 10}, {3, 10}}, 
X   {{2, 11}, {3, 3}}, {{2, 11}, {3, 11}}, 
X   {{2, 11}, {3, 12}}, {{2, 12}, {3, 1}}, 
X   {{2, 12}, {3, 11}}, {{2, 12}, {3, 13}}, 
X   {{2, 13}, {3, 2}}, {{2, 13}, {3, 12}}, 
X   {{2, 13}, {3, 13}}, {{2, 14}, {3, 5}}, 
X   {{2, 14}, {3, 11}}, {{2, 14}, {3, 14}}, 
X   {{2, 15}, {3, 6}}, {{2, 15}, {3, 12}}, 
X   {{2, 15}, {3, 14}}, {{2, 16}, {3, 4}}, 
X   {{2, 16}, {3, 13}}, {{2, 16}, {3, 14}}, 
X   {{2, 17}, {3, 8}}, {{2, 17}, {3, 11}}, 
X   {{2, 17}, {3, 15}}, {{2, 18}, {3, 9}}, 
X   {{2, 18}, {3, 12}}, {{2, 18}, {3, 15}}, 
X   {{2, 19}, {3, 7}}, {{2, 19}, {3, 13}}, 
X   {{2, 19}, {3, 15}}, {{2, 20}, {3, 10}}, 
X   {{2, 20}, {3, 16}}, {{2, 20}, {3, 17}}, 
X   {{2, 21}, {3, 4}}, {{2, 21}, {3, 16}}, 
X   {{2, 21}, {3, 18}}, {{2, 22}, {3, 7}}, 
X   {{2, 22}, {3, 17}}, {{2, 22}, {3, 18}}, 
X   {{2, 23}, {3, 5}}, {{2, 23}, {3, 16}}, 
X   {{2, 23}, {3, 19}}, {{2, 24}, {3, 8}}, 
X   {{2, 24}, {3, 17}}, {{2, 24}, {3, 19}}, 
X   {{2, 25}, {3, 1}}, {{2, 25}, {3, 18}}, 
X   {{2, 25}, {3, 19}}, {{2, 26}, {3, 6}}, 
X   {{2, 26}, {3, 16}}, {{2, 26}, {3, 20}}, 
X   {{2, 27}, {3, 9}}, {{2, 27}, {3, 17}}, 
X   {{2, 27}, {3, 20}}, {{2, 28}, {3, 2}}, 
X   {{2, 28}, {3, 18}}, {{2, 28}, {3, 20}}, 
X   {{2, 29}, {3, 3}}, {{2, 29}, {3, 19}}, 
X   {{2, 29}, {3, 20}}, {{2, 30}, {3, 11}}, 
X   {{2, 30}, {3, 19}}, {{2, 30}, {3, 21}}, 
X   {{2, 31}, {3, 12}}, {{2, 31}, {3, 20}}, 
X   {{2, 31}, {3, 21}}, {{2, 32}, {3, 14}}, 
X   {{2, 32}, {3, 16}}, {{2, 32}, {3, 21}}, 
X   {{2, 33}, {3, 15}}, {{2, 33}, {3, 17}}, 
X   {{2, 33}, {3, 21}}, {{2, 34}, {3, 13}}, 
X   {{2, 34}, {3, 18}}, {{2, 34}, {3, 21}}, 
X   {{3, 1}, {4, 1}}, {{3, 1}, {4, 2}}, 
X   {{3, 2}, {4, 1}}, {{3, 2}, {4, 3}}, 
X   {{3, 3}, {4, 2}}, {{3, 3}, {4, 3}}, 
X   {{3, 4}, {4, 1}}, {{3, 4}, {4, 4}}, 
X   {{3, 5}, {4, 2}}, {{3, 5}, {4, 4}}, 
X   {{3, 6}, {4, 3}}, {{3, 6}, {4, 4}}, 
X   {{3, 7}, {4, 1}}, {{3, 7}, {4, 5}}, 
X   {{3, 8}, {4, 2}}, {{3, 8}, {4, 5}}, 
X   {{3, 9}, {4, 3}}, {{3, 9}, {4, 5}}, 
X   {{3, 10}, {4, 4}}, {{3, 10}, {4, 5}}, 
X   {{3, 11}, {4, 2}}, {{3, 11}, {4, 6}}, 
X   {{3, 12}, {4, 3}}, {{3, 12}, {4, 6}}, 
X   {{3, 13}, {4, 1}}, {{3, 13}, {4, 6}}, 
X   {{3, 14}, {4, 4}}, {{3, 14}, {4, 6}}, 
X   {{3, 15}, {4, 5}}, {{3, 15}, {4, 6}}, 
X   {{3, 16}, {4, 4}}, {{3, 16}, {4, 7}}, 
X   {{3, 17}, {4, 5}}, {{3, 17}, {4, 7}}, 
X   {{3, 18}, {4, 1}}, {{3, 18}, {4, 7}}, 
X   {{3, 19}, {4, 2}}, {{3, 19}, {4, 7}}, 
X   {{3, 20}, {4, 3}}, {{3, 20}, {4, 7}}, 
X   {{3, 21}, {4, 6}}, {{3, 21}, {4, 7}}, 
X   {{4, 1}, {5, 1}}, {{4, 2}, {5, 1}}, 
X   {{4, 3}, {5, 1}}, {{4, 4}, {5, 1}}, 
X   {{4, 5}, {5, 1}}, {{4, 6}, {5, 1}}, {{4, 7}, {5, 1}}}
X   ]
X;[o]
X{{{-1, 1}, {0, 1}}, <<341>>, {{4, 7}, {5, 1}}}
X:[font = text; inactive; preserveAspect; ]
XHere, each element {{k,m},{k+1,n}} of flinks indicates that  the m-th face of dimension k is  covered by the n-th face of dimension k+1.
X:[font = text; inactive; preserveAspect; ]
XSince we have imported Combinatorica package also,  we conlude this notebook with explanation for drawing the graph of the polytope using the package:
X:[font = input; dontPreserveAspect; ]
Xedges=EdgesOfPolyhedron[activesets];
X:[font = input; dontPreserveAspect; startGroup; ]
XShowLabeledGraph[g=FromUnorderedPairs[edges]]
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X%!
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X[ 1 1 0 0 ]
X] MathScale
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X%%Object: Graphics
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END_OF_FILE
if test 48676 -ne `wc -c <'FaceLattice-notebook.ma'`; then
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fi
# end of 'FaceLattice-notebook.ma'
fi
echo shar: End of shell archive.
exit 0
