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The complexity of facets (and some facets of complexity)

Published:05 May 1982Publication History

ABSTRACT

Many important combinatorial optimization problems, including the traveling salesman problem (TSP), the clique problem and many others, call for the optimization of a linear functional over some discrete set of vectors.

References

  1. 1.E. Balas , "Facets of the knapsack polytope", Math.Progr. 8, 146-164, (1975).Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. 2.E. Balas and E. Zemel, "Critical cutsets of graphs and canonical facets of set packing polytopes", Math. of Oper. Res. 2, 15-20, (1977).Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. 3.C. Berge, Graphs and Hypergraphs, North-Holland, 1973. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. 4.V. Chvatal, "Edmonds polytopes and weakly hamiltonian graphs", Math.Progr. 5, 29-40, (1973).Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. 5.V. Chvatal, "On certain polytopes associated with graphs", J. of Comb. Theory (B) 18, 138-154, (1975).Google ScholarGoogle ScholarCross RefCross Ref
  6. 6.G. Dantzig, Linear Programming and Extensions, Princeton Univ. Press, 1963.Google ScholarGoogle Scholar
  7. 7.G. B. Dantzig, D. R. Fulkerson, S. M. Johnson, "Solution of a large-scale travelling salesman problem", Operations Research 2, 393-410, (1954).Google ScholarGoogle ScholarCross RefCross Ref
  8. 8.J. Doyen and V. van Diest, "New families of hypohamiltonian graphs", Discrete Math. 13, 225-236, (1975).Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. 9.M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completenes, W. H. Freeman, 1979. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. 10.M. Grotschel, "On the monotone symmetric travelling salesman problem:hypohamiltonian/hypotraceable graphs and facets", Math. of Oper. Res. 5, 285-292, (1980).Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. 11.M. Grotschel, L. Lovasz, S. Schrijver, "The ellipsoid method and its consequences in combinatorial optimization", Bonn, (1980).Google ScholarGoogle Scholar
  12. 12.M. Grotschel and M. W, Padberg, "On the symmetric travelling salesman problem II: Lifting theorem and facets", Math. Progr. 16, 281-302, (1979).Google ScholarGoogle ScholarCross RefCross Ref
  13. 13.F. Harary, Graph Theory, Addison Wesley, 1969.Google ScholarGoogle Scholar
  14. 14.R. M. Karp, "Reducibility among combinatorial problems", in Complexity of Computer Computations, R. E. Miller and J. W. Thatcher eds., Plenum Press, 85-103, (1972).Google ScholarGoogle Scholar
  15. 15.R. M. Karp and C. H. Papadimitriou, "On linear characterizations of combinatorial optimization problems", Proc. 21st FOCS, 1-9, (1980).Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. 16.D. Kelly, "The 3-irreducible partially ordered sets", Can. J. Math. 29, 367-383, (1977).Google ScholarGoogle Scholar
  17. 17.L. G. Khacian, "A polynomial algorithm for linear programming", Dokl. Ak. Nauk. SSSR, 1093-1096, (1979).Google ScholarGoogle Scholar
  18. 18.E. W. Leggett, Jr. and D. J. Moore, "Optimization problems and the polynomial hierarchy", Theor. Comp. Science 15, 279-289, (1981).Google ScholarGoogle ScholarCross RefCross Ref
  19. 19.W. F. Lindgren, "An infinite class of hypohamiltonian graphs", Amer. Math. Monthly 74, 1087-1089, (1967).Google ScholarGoogle ScholarCross RefCross Ref
  20. 20.G. L. Nemhauser and L. E. Trotter, "Properties of vertex packing and independence system polyhedra", Math. Progr. 6, 48-61, (1974).Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. 21.M. W. Padberg, "On the facial structure of set packing polyhedra", Math. Progr. 5, 199-215, (1973).Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. 22.M. W. Padberg, "On the complexity of set packing polyhedra", Annals of Discrete Math. 1, 421-434, (1977).Google ScholarGoogle ScholarCross RefCross Ref
  23. 23.C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, 1982. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. 24.C. Thomassen, "Hypohamiltonian and hypotraceable graphs", Discrete Math. 9, 91-96, (1974).Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. 25.W. T. Trotter Jr. and J. I. Moore Jr., "Characterization problems for graphs, partially ordered sets, lattices and families of sets", Discrete Math. 16, 361-381, (1976).Google ScholarGoogle ScholarCross RefCross Ref
  26. 26.W. T. Trotter Jr. and J. A. Ross, "For t-&-ge;3, every t-dimensonal partial order can be embedded in a t+1-irreducible partial order", (1981).Google ScholarGoogle Scholar
  27. 27.L. A. Wolsey, "Further facet generating procedures for vertex packing polytopes", Math. Progr. 11, 158-163, (1976).Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. 28.M. Yannakakis, "The complexity of the partial order dimension problem", SIAM J. Appl. Dis. Meth., to appear.Google ScholarGoogle Scholar
  1. The complexity of facets (and some facets of complexity)

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            cover image ACM Conferences
            STOC '82: Proceedings of the fourteenth annual ACM symposium on Theory of computing
            May 1982
            408 pages
            ISBN:0897910702
            DOI:10.1145/800070

            Copyright © 1982 ACM

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            • Published: 5 May 1982

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