(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 116495, 3064]*) (*NotebookOutlinePosition[ 117550, 3100]*) (* CellTagsIndexPosition[ 117506, 3096]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Generating All Faces of Convex Polytopes:FaceLattice.m\n-", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" Package -\nVersion 0.2 Beta, September 19, 1992", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12], StyleBox[ "\nKomei Fukuda(University of Tsukuba, Tokyo) \nand \nVera Rosta(Temple \ University Japan)", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "In 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 ", Evaluatable->False, AspectRatioFixed->True], StyleBox["bounded", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " 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 ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" package which is to be in the directory DiscreteMath. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["$Path=Join[$Path, {\"~/NeXT/NeXTmath/VertexEnum\"}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {\".\", \"~\", \"~/Library/Mathematica/Packages\", \"/LocalApps/Mathematica.app/Kernel/NeXT/Preload\", \"/LocalApps/Mathematica.app/Kernel/NeXT/StartUp\", \"/LocalLibrary/Mathematica/Packages\", \"~/NeXT/NeXTmath/VertexEnum\"}\ \>", "\<\ {., ~, ~/Library/Mathematica/Packages, /LocalApps/Mathematica.app/Kernel/NeXT/Preload, /LocalApps/Mathematica.app/Kernel/NeXT/StartUp, /LocalLibrary/Mathematica/Packages, ~/NeXT/NeXTmath/VertexEnum}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["<True, AspectRatioFixed->True], Cell[TextData["Off[General::spell]; Off[General::spell1]; "], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Let 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."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["size=2; dim=5;"], "Input", AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData["m=Table[Table[Random[Integer,{1,5}],{dim}],{size}]"], "Input", AspectRatioFixed->False], Cell[OutputFormData["\<\ {{5, 1, 4, 5, 1}, {4, 2, 1, 2, 1}}\ \>", "\<\ {{5, 1, 4, 5, 1}, {4, 2, 1, 2, 1}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->False]}, Open]], Cell[CellGroupData[{Cell[TextData["b=Table[Sum[m[[i,j]],{j,dim}],{i,size}]"], "Input", AspectRatioFixed->False], Cell[OutputFormData["\<\ {16, 10}\ \>", "\<\ {16, 10}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->False]}, Open]]}, Open]], Cell[TextData[ "Then we compute all vertices of the polytope using the VetexEnumeration \ function of VertexEnum package. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["vlist=VertexEnumeration[m,b];"], "Input", AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData["activesets=vlist[[2]]"], "Input", AspectRatioFixed->False], Cell[OutputFormData["\<\ {{1, 2, 3, 4, 5}, {1, 2, 3, 6, 5}, {7, 2, 3, 4, 5}, {7, 2, 3, 6, 5}, {7, 2, 4, 6, 5}, {1, 2, 4, 6, 5}, {7, 1, 4, 6, 5}, {7, 2, 4, 6, 1}, {7, 1, 3, 6, 5}, {7, 2, 3, 6, 1}, {7, 1, 3, 4, 5}, {7, 2, 3, 4, 1}}\ \>", "\<\ {{1, 2, 3, 4, 5}, {1, 2, 3, 6, 5}, {7, 2, 3, 4, 5}, {7, 2, 3, 6, 5}, {7, 2, 4, 6, 5}, {1, 2, 4, 6, 5}, {7, 1, 4, 6, 5}, {7, 2, 4, 6, 1}, {7, 1, 3, 6, 5}, {7, 2, 3, 6, 1}, {7, 1, 3, 4, 5}, {7, 2, 3, 4, 1}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["Here ", Evaluatable->False, AspectRatioFixed->True], StyleBox["activesets", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " is the list of vertices, each represented as the set of indices of \ inequalities which are satisfied by equality (i.e. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["active", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ ") 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 ", Evaluatable->False, AspectRatioFixed->True], StyleBox["KFaceList", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " 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.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(* The following takes about two minutes on Macintosh SE/30 *)"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["flist=KFaceList[activesets] "], "Input", AspectRatioFixed->True], Cell[TextData[ "All 0-faces determined.\nAll 1-faces determined.\nAll 2-faces determined.\n\ All 3-faces determined.\nAll 4-faces determined."], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ {{{1, 2, 3, 4, 5, 6, 7}}, {{1, 2, 3, 4, 5}, {1, 2, 3, 5, 6}, {2, 3, 4, 5, 7}, {2, 3, 5, 6, 7}, {2, 4, 5, 6, 7}, {1, 2, 4, 5, 6}, {1, 4, 5, 6, 7}, {1, 2, 4, 6, 7}, {1, 3, 5, 6, 7}, {1, 2, 3, 6, 7}, {1, 3, 4, 5, 7}, {1, 2, 3, 4, 7}}, {{1, 2, 3, 5}, {2, 3, 4, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}, {1, 2, 3, 4}, {2, 3, 5, 6}, {1, 2, 5, 6}, {1, 3, 5, 6}, {1, 2, 3, 6}, {2, 3, 5, 7}, {2, 4, 5, 7}, {3, 4, 5, 7}, {2, 3, 4, 7}, {2, 5, 6, 7}, {3, 5, 6, 7}, {2, 3, 6, 7}, {2, 4, 5, 6}, {4, 5, 6, 7}, {2, 4, 6, 7}, {1, 4, 5, 6}, {1, 2, 4, 6}, {1, 4, 6, 7}, {1, 5, 6, 7}, {1, 4, 5, 7}, {1, 2, 6, 7}, {1, 2, 4, 7}, {1, 3, 6, 7}, {1, 3, 5, 7}, {1, 2, 3, 7}, {1, 3, 4, 7}}, {{2, 3, 5}, {1, 2, 5}, {1, 3, 5}, {1, 2, 3}, {2, 4, 5}, {3, 4, 5}, {2, 3, 4}, {1, 4, 5}, {1, 2, 4}, {1, 3, 4}, {2, 5, 6}, {3, 5, 6}, {2, 3, 6}, {1, 5, 6}, {1, 2, 6}, {1, 3, 6}, {2, 5, 7}, {3, 5, 7}, {2, 3, 7}, {4, 5, 7}, {2, 4, 7}, {3, 4, 7}, {5, 6, 7}, {2, 6, 7}, {3, 6, 7}, {4, 5, 6}, {2, 4, 6}, {4, 6, 7}, {1, 4, 6}, {1, 6, 7}, {1, 4, 7}, {1, 5, 7}, {1, 2, 7}, {1, 3, 7}}, {{2, 5}, {3, 5}, {2, 3}, {1, 5}, {1, 2}, {1, 3}, {4, 5}, {2, 4}, {3, 4}, {1, 4}, {5, 6}, {2, 6}, {3, 6}, {1, 6}, {5, 7}, {2, 7}, {3, 7}, {4, 7}, {6, 7}, {4, 6}, {1, 7}}, {{5}, {2}, {3}, {1}, {4}, {6}, {7}}, {{}}}\ \>", "\<\ {{{1, 2, 3, 4, 5, 6, 7}}, {{1, 2, 3, 4, 5}, {1, 2, 3, 5, 6}, {2, 3, 4, 5, 7}, {2, 3, 5, 6, 7}, {2, 4, 5, 6, 7}, {1, 2, 4, 5, 6}, {1, 4, 5, 6, 7}, {1, 2, 4, 6, 7}, {1, 3, 5, 6, 7}, {1, 2, 3, 6, 7}, {1, 3, 4, 5, 7}, {1, 2, 3, 4, 7}}, {{1, 2, 3, 5}, {2, 3, 4, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}, {1, 2, 3, 4}, {2, 3, 5, 6}, {1, 2, 5, 6}, {1, 3, 5, 6}, {1, 2, 3, 6}, {2, 3, 5, 7}, {2, 4, 5, 7}, {3, 4, 5, 7}, {2, 3, 4, 7}, {2, 5, 6, 7}, {3, 5, 6, 7}, {2, 3, 6, 7}, {2, 4, 5, 6}, {4, 5, 6, 7}, {2, 4, 6, 7}, {1, 4, 5, 6}, {1, 2, 4, 6}, {1, 4, 6, 7}, {1, 5, 6, 7}, {1, 4, 5, 7}, {1, 2, 6, 7}, {1, 2, 4, 7}, {1, 3, 6, 7}, {1, 3, 5, 7}, {1, 2, 3, 7}, {1, 3, 4, 7}}, {{2, 3, 5}, {1, 2, 5}, {1, 3, 5}, {1, 2, 3}, {2, 4, 5}, {3, 4, 5}, {2, 3, 4}, {1, 4, 5}, {1, 2, 4}, {1, 3, 4}, {2, 5, 6}, {3, 5, 6}, {2, 3, 6}, {1, 5, 6}, {1, 2, 6}, {1, 3, 6}, {2, 5, 7}, {3, 5, 7}, {2, 3, 7}, {4, 5, 7}, {2, 4, 7}, {3, 4, 7}, {5, 6, 7}, {2, 6, 7}, {3, 6, 7}, {4, 5, 6}, {2, 4, 6}, {4, 6, 7}, {1, 4, 6}, {1, 6, 7}, {1, 4, 7}, {1, 5, 7}, {1, 2, 7}, {1, 3, 7}}, {{2, 5}, {3, 5}, {2, 3}, {1, 5}, {1, 2}, {1, 3}, {4, 5}, {2, 4}, {3, 4}, {1, 4}, {5, 6}, {2, 6}, {3, 6}, {1, 6}, {5, 7}, {2, 7}, {3, 7}, {4, 7}, {6, 7}, {4, 6}, {1, 7}}, {{5}, {2}, {3}, {1}, {4}, {6}, {7}}, {{}}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "The function FVector counts the number of k-faces for k=-1,0,2,...,d."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["fvector=FVector[flist]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {1, 12, 30, 34, 21, 7, 1}\ \>", "\<\ {1, 12, 30, 34, 21, 7, 1}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "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. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["avector=Table[(-1)^i,{i,Length[flist]}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {-1, 1, -1, 1, -1, 1, -1}\ \>", "\<\ {-1, 1, -1, 1, -1, 1, -1}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "The following must be true. 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