Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube

  • Friedrich Eisenbrand
  • Andreas S. Schulz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1610)

Abstract

Gomory’s and Chvátal’s cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvátal rank of the polyhedron. It is well-known that the Chvátal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1-polytope. We prove that the Chvátal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, the rank of any polytope contained in the n-dimensional 0/1-cube is at most 3n2 lg n. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n3 lg n).

Moreover, we refine this result by showing that the rank of any polytope in the 0/1-cube that is defined by inequalities with small coefficients is O(n). The latter observation explains why for most cutting planes derived in polyhedral studies of several popular combinatorial optimization problems only linear growth has been observed (see, e.g., [13]); the coefficients of the corresponding inequalities are usually small. Similar results were only known for monotone polyhedra before.

Finally, we provide a family of polytopes contained in the 0/1-cube the Chvátal rank of which is at least (1+)n for some > 0; the best known lower bound was n.

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References

  1. [1]
    N. Alon and V. H. Vu. Anti-Hadamard matrices, coin weighing, threshold gates, and indecomposable hypergraphs. Journal of Combinatorial Theory, 79A:133–160, 1997.CrossRefMathSciNetGoogle Scholar
  2. [2]
    E. Balas, S. Ceria, G. Cornuéjols, and N. R. Natraj. Gomory cuts revisited. Operations Research Letters, 19:1–9, 1996.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    E. Balas and M. J. Saltzman. Facets of the three-index assignment polytope. Discrete Applied Mathematics, 23:201–229, 1989.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    F. Barahona, M. Grötschel, and A. R. Mahjoub. Facets of the bipartite subgraph polytope. Mathematics of Operations Research, 10:340–358, 1985.MATHMathSciNetGoogle Scholar
  5. [5]
    A. Bockmayr and F. Eisenbrand. On the Chvátal rank of polytopes in the 0/1 cube. Research Report MPI-I-97-2-009, Max-Planck-Institut für Informatik, September 1997.Google Scholar
  6. [6]
    A. Bockmayr, F. Eisenbrand, M. E. Hartmann, and A. S. Schulz. On the Chvátal rank of polytopes in the 0/1 cube. Technical Report 616, Technical University of Berlin, Department of Mathematics, December 1998.Google Scholar
  7. [7]
    M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. Journal of Symbolic Logic, 62:708–728, 1997.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    S. C. Boyd and W. H. Cunningham. Small travelling salesman polytopes. Mathematics of Operations Research, 16:259–271, 1991.MATHMathSciNetGoogle Scholar
  9. [9]
    S. C. Boyd, W. H. Cunningham, M. Queyranne, and Y. Wang. Ladders for travelling salesmen. SIAM Journal on Optimization, 5:408–420, 1995.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    S. C. Boyd and W. R. Pulleyblank. Optimizing over the subtour polytope of the travelling salesman problem. Mathematical Programming, 49:163–187, 1991.CrossRefMathSciNetGoogle Scholar
  11. [11]
    V. Chvátal. Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Mathematics, 4:305–337, 1973.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    V. Chvátal. Flip-flops in hypohamiltonian graphs. Canadian Mathematical Bulletin, 16:33–41, 1973.MATHMathSciNetGoogle Scholar
  13. [13]
    V. Chvátal, W. Cook, and M. E. Hartmann. On cutting-plane proofs in combinatorial optimization. Linear Algebra and its Applications, 114/115:455–499, 1989.CrossRefGoogle Scholar
  14. [14]
    W. Cook, C. R. Coullard, and Gy. Turán. On the complexity of cutting plane proofs. Discrete Applied Mathematics, 18:25–38, 1987.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    W. Cook, W. H. Cunningham, W. R. Pulleyblank, and A. Schrijver. Combinatorial Optimization. John Wiley, 1998.Google Scholar
  16. [16]
    F. Eisenbrand. A note on the membership problem for the first elementary closure of a polyhedron. Technical Report 605, Technical University of Berlin, Department of Mathematics, November 1998. To appear in Combinatorica.Google Scholar
  17. [17]
    P. Erdös. On circuits and subgraphs of chromatic graphs. Mathematika, 9:170–175, 1962.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    M. Fischetti. Three facet lifting theorems for the asymmetric traveling salesman polytope. In E. Balas, G. Cournuéjols, and R. Kannan, editors, Integer Programming and Combinatorial Optimization, pages 260–273. Proceedings of the 2nd IPCO Conference, 1992.Google Scholar
  19. [19]
    T. Fleiner, V. Kaibel, and G. Rote. Upper bounds on the maximal number of facets of 0/1-polytopes. Technical Report 98-327, University of Cologne, Department of Computer Science, 1998. To appear in European Journal of Combinatorics.Google Scholar
  20. [20]
    R. Giles and L. E. Trotter. On stable set polyhedra for K 1,3-free graphs. Journal of Combinatorial Theory, 31:313–326, 1981.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    R. E. Gomory. Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society, 64:275–278, 1958.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    R. E. Gomory. An algorithm for integer solutions to linear programs. In R. L. Graves and P. Wolfe, editors, Recent Advances in Mathematical Programming, pages 269–302. McGraw-Hill, 1963.Google Scholar
  23. [23]
    M. Grötschel and M. W. Padberg. Polyhedral theory. In E. L. Lawler, J. K. Lenstra, A. H. G. Rinnoy Kan, and D. B. Shmoys, editors, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, pages 251–305. John Wiley, 1985.Google Scholar
  24. [24]
    M. Grötschel and W. R. Pulleyblank. Clique tree inequalities and the symmetric travelling salesman problem. Mathematics of Operations Research, 11:537–569, 1986.MATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    A. Haken. The intractability of resolution. Theoretical Computer Science, 39:297–308, 1985.MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    P. L. Hammer, E. Johnson, and U. N. Peled. Facets of regular 0–1 polytopes. Mathematical Programming, 8:179–206, 1975.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    M. E. Hartmann. Cutting planes and the complexity of the integer hull. Technical Report 819, School of Operations Research and Industrial Engineering, Cornell University, September 1988.Google Scholar
  28. [28]
    M. E. Hartmann. Personal communication, March 1998.Google Scholar
  29. [29]
    M. E. Hartmann, M. Queyranne, and Y. Wang. On the Chvátal rank of certain inequalities. This volume, 1999.Google Scholar
  30. [30]
    R. Impagliazzo, T. Pitassi, and A. Urquhart. Upper and lower bound for tree-like cutting plane proofs. In Proc. Logic in Computer Science, LICS’94, Paris, 1994.Google Scholar
  31. [31]
    U. H. Kortenkamp, J. Richter-Gebert, A. Sarangarajan, and G. M. Ziegler. Extremal properties of 0/1-polytopes. Discrete and Computational Geometry, 17:439–448, 1997.MATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    D. Naddef. The Hirsch conjecture is true for (0,1)-polytopes. Mathematical Programming, 45:109–110, 1989.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    M. W. Padberg and M. Grötschel. Polyhedral computations. In E. L. Lawler, J. K. Lenstra, A. H. G. Rinnoy Kan, and D. B. Shmoys, editors, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, pages 307–360. John Wiley, 1985.Google Scholar
  34. [34]
    P. Pudlák. Lower bounds for resolution and cutting plane proofs and monotone computations. Journal of Symbolic Logic, 62:981–988, 1997.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    W. R. Pulleyblank. Polyhedral combinatorics. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization, Volume 1 of Handbooks in Operations Research and Management Science, Chapter V, pages 371–446. Elsevier, 1989.Google Scholar
  36. [36]
    A. Schrijver. On cutting planes. Annals of Discrete Mathematics, 9:291–296, 1980.MATHMathSciNetCrossRefGoogle Scholar
  37. [37]
    A. Schrijver. Theory of Linear and Integer Programming. John Wiley, 1986.Google Scholar
  38. [38]
    A. S. Schulz. Polytopes and Scheduling. PhD thesis, Technical University of Berlin, Berlin, Germany, 1996.MATHGoogle Scholar
  39. [39]
    A. S. Schulz. A simple proof that the Chvátal rank of polytopes in the 0/1-cube is small. Unpublished manuscript, September 1997.Google Scholar
  40. [40]
    A. S. Schulz, R. Weismantel, and G. M. Ziegler. An optimization problem is ten problems. In preparation.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Friedrich Eisenbrand
    • 1
  • Andreas S. Schulz
    • 2
  1. 1.Max-Planck-Institut für InformatikIm StadtwaldSaarbrückenGermany
  2. 2.Sloan School of Management and Operations Research Center, E53-361MITCambridgeUSA

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