Cones of Matrices and Set-Functions and 0–1 Optimization

It has been recognized recently that to represent a polyhedron as the projection of a higher-dimensional, but simpler, polyhedron, is a powerful tool in polyhedral combinatorics. A general method is developed to construct higher-dimensional polyhedra (or, in some cases, convex sets) whose projection approximates the convex hull of 0–1 valued solutions of a system of linear inequalities. An important feature of these approximations is that one can optimize any linear objective function over them in polynomial time.

In the special case of the vertex packing polytope, a sequence of systems of inequalities is obtained such that the first system already includes clique, odd hole, odd antihole, wheel, and orthogonality constraints. In particular, for perfect (and many other) graphs, this first system gives the vertex packing polytope. For various classes of graphs, including t-perfect graphs, it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets.

An extension of the method is also discussed which establishes a connection with certain submodular functions and the Möbius function of a lattice.

  • [1]  E. Balas and , W. R. Pulleyblank, The perfectly matchable subgraph polytope of a bipartite graph, Networks, 13 (1983), 495–516 85d:05189 0525.90069 CrossrefISIGoogle Scholar

  • [2]  E. Balas and , W. R. Pulleyblank, The perfectly matchable subgraph polytope of an arbitrary graph, Combinatorica, 9 (1989), 321–337 91e:05060 0723.05087 CrossrefISIGoogle Scholar

  • [3]  M. O. Ball, W. Liu and , W. R. Pulleyblank, B. Tulkens and , H. Tulkens, Two-terminal Steiner tree polyhedraContributions to operations research and economics (Louvain-la-Neuve, 1987), MIT Press, Cambridge, MA, 1989, 251–284 92h:90128 Google Scholar

  • [4]  F. Barahona, Reducing matching to polynomial size linear programming, Res. Report, CORR 88-51, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, 1988 Google Scholar

  • [5]  F. Barahona and , A. R. Mahjoub, Compositions of graphs and polyhedra II: Stable sets, Res. Report, 87464-OR, Institut für Operations Research, Universität Bonn , Bonn, FRG, 1987 Google Scholar

  • [6]  Kathie Cameron and , Jack Edmonds, Coflow polyhedra, Discrete Math., 101 (1992), 1–21 10.1016/0012-365X(92)90585-4 93d:90040 0773.90025 CrossrefISIGoogle Scholar

  • [7]  S. Ceria, 1989, personal communication Google Scholar

  • [8]  V. Chvátal, Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Math., 4 (1973), 305–337 47:1635 0253.05131 CrossrefGoogle Scholar

  • [9]  V. Chvátal, On certain polytopes associated with graphs, J. Combinatorial Theory Ser. B, 18 (1975), 138–154 51:7949 0277.05139 CrossrefISIGoogle Scholar

  • [10]  Jack Edmonds, Maximum matching and a polyhedron with $0,1$-vertices, J. Res. Nat. Bur. Standards Sect. B, 69B (1965), 125–130 32:1012 0141.21802 CrossrefISIGoogle Scholar

  • [11]  Ralph E. Gomory, R. Graves and , P. Wolfe, An algorithm for integer solutions to linear programsRecent advances in mathematical programming, McGraw-Hill, New York, 1963, 269–302 30:4594 0235.90038 Google Scholar

  • [12]  M. Grötschel, L. Lovász and , A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica, 1 (1981), 169–197 84a:90044 0557.10036 CrossrefISIGoogle Scholar

  • [13]  M. Grötschel, L. Lovász and , A. Schrijver, Relaxations of vertex packing, J. Combin. Theory Ser. B, 40 (1986), 330–343 87h:05087 0596.05052 CrossrefISIGoogle Scholar

  • [14]  Martin Grötschel, László Lovász and , Alexander Schrijver, Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics: Study and Research Texts, Vol. 2, Springer-Verlag, Berlin, 1988xii+362, New York 89m:90135 CrossrefGoogle Scholar

  • [15]  Bernt Lindström, Determinants on semilattices, Proc. Amer. Math. Soc., 20 (1969), 207–208 39:102 0165.02902 CrossrefISIGoogle Scholar

  • [16]  W. Liu, Ph.D. Thesis, Extended formulations and polyhedral projection, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, 1988 Google Scholar

  • [17]  L. Lovász, Combinatorial problems and exercises, North-Holland Publishing Co., Amsterdam, 1979, 551–, Akadémiai Kiadó, Budapest, Hungary; the Netherlands 80m:05001 0439.05001 Google Scholar

  • [18]  L. Lovász and , M. D. Plummer, Matching theory, North-Holland Mathematics Studies, Vol. 121, North-Holland Publishing Co., Amsterdam, 1986xxvii+544, Elsevier, the Netherlands 88b:90087 0618.05001 Google Scholar

  • [19]  N. Maculan, The Steiner problem in graphs, Ann Discrete Math., 31 (1987), 185–222 0622.90029 Google Scholar

  • [20]  Robin Pemantle, James Propp and , Daniel Ullman, On tensor powers of integer programs, SIAM J. Discrete Math., 5 (1992), 127–143 10.1137/0405011 93b:90052 0751.90053 LinkISIGoogle Scholar

  • [21]  Gian-Carlo Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 2 (1964), 340–368 (1964) 30:4688 0121.02406 CrossrefGoogle Scholar

  • [22]  Hanif D. Sherali and , Warren P. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM J. Discrete Math., 3 (1990), 411–430 10.1137/0403036 91k:90116 0712.90050 LinkISIGoogle Scholar

  • [23]  Richard P. Stanley, Enumerative combinatorics. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986xiv+306 87j:05003 0608.05001 CrossrefGoogle Scholar

  • [24]  Herbert S. Wilf, Hadamard determinants, Möbius functions, and the chromatic number of a graph, Bull. Amer. Math. Soc., 74 (1968), 960–964 37:5106 0172.01602 CrossrefISIGoogle Scholar

  • [25]  R. J. Wilson, P. Erdös, A. Rényi and , V. T. Sós, The Selberg sieve for a latticeCombinatorial theory and its applications, III (Proc. Colloq., Balatonfüred, 1969), Vol. 4, North-Holland, Amsterdam, 1970, 1141–1149, Coll. Math. Soc. János Bolyai 47:167 0213.33203 Google Scholar

  • [26]  M. Yannakakis, Expressing combinatorial optimization problems by linear programs, Proc. 29th IEEE Symposium on Foundations of Computer Science, White Plains, NY, 1988, 223–228 Google Scholar