The ellipsoid method and its consequences in combinatorial optimization

Abstract

L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields a proof that weighted fractional chromatic number is NP-hard.

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References

  1. [1965]

    Chu Yoeng-jin andLiu Tseng-hung, On the shortest arborescence of a directed graph,Scientia Sinica 4 (1965) 1396–1400.

    Google Scholar 

  2. [1959]

    E. W. Dijkstra, A note on two problems in connexion with graphs,Numer. Math. 1 (1959) 269–271.

    MATH  Article  MathSciNet  Google Scholar 

  3. [1965]

    J. Edmonds, Maximum matching and a polyhedron with 0, 1-vertices,J. Res. Nat. Bur. Standards Sect. B 69 (1965) 125–130.

    MATH  MathSciNet  Google Scholar 

  4. [1967]

    J. Edmonds, Optimum branchings,J. Res. Nat. Bur. Standards Sect. B 71 (1967) 233–240.

    MATH  MathSciNet  Google Scholar 

  5. [1970]

    J. Edmonds, Submodular functions, matroids, and certain polyhedra, in:Combinatorial structures and their applications, Proc. Intern. Conf. Calgary, Alb., 1969 (R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds.), Gordon and Breach, New York, 1970, 69–87.

    Google Scholar 

  6. [1970a]

    J. Edmonds andE. L. Johnson, Matching: a well-solved class of integer linear programs, —ibid. 89–92.

    Google Scholar 

  7. [1973]

    J. Edmonds, Edge-disjoint branchings, in:Combinatorial algorithms, Courant Comp. Sci. Symp. Monterey, Ca., 1972 (R. Rustin, ed.), Acad. Press, New York, 1973, 91–96.

    Google Scholar 

  8. [1979]

    J. Edmonds, Matroid intersection,Annals of Discrete Math. 4 (1979) 39–49.

    MATH  MathSciNet  Google Scholar 

  9. [1977]

    J. Edmonds andR. Giles, A min-max relation for submodular functions on graphs,Annals of Discrete Math. 1 (1977) 185–204.

    MathSciNet  Article  Google Scholar 

  10. [1973]

    J. Edmonds andE. L. Johnson, Matching, Euler tours and the Chinese postman,Math. Programming 5 (1973) 88–124.

    MATH  Article  MathSciNet  Google Scholar 

  11. [1956]

    L. R. Ford andD. R. Fulkerson, Maximum flow through a network,Canad. J. Math. 8 (1956) 399–404.

    MATH  MathSciNet  Google Scholar 

  12. [1962]

    L. R. Ford andD. R. Fulkerson,Flows in networks, Princeton Univ. Press, Princeton, N. J., 1962.

    Google Scholar 

  13. [1980]

    A. Frank, On the orientations of graphs,J. Combinatorial Theory (B) 28 (1980) 251–260.

    MATH  Article  Google Scholar 

  14. [1979]

    A. Frank, Kernel systems of directed graphs,Acta Sci. Math. (Szeged)41 (1979) 63–76.

    MATH  MathSciNet  Google Scholar 

  15. [1981]

    A. Frank, How to make a digraph strongly connected,Combinatorica 1 (2) (1981) 145–153.

    MATH  MathSciNet  Google Scholar 

  16. [1968]

    D. R. Fulkerson, Networks, frames, and blocking systems, in:Mathematics of the decision sciences, Part I (G. B. Dantzig and A. F. Veinott, eds.), Amer. Math. Soc., Providence, R. I., 1968, 303–334.

    Google Scholar 

  17. [1970b]

    D. R. Fulkerson, Blocking polyhedra, in:Graph theory and its applications, Proc. adv. Seminar Madison, Wis., 1969 (B. Harris, ed.), Acad. Press, New York, 1970, 93–112.

    Google Scholar 

  18. [1974]

    D. R. Fulkerson, Packing rooted directed cuts in a weighted directed graph,Math. Programming 6 (1974) 1–13.

    MATH  Article  MathSciNet  Google Scholar 

  19. [1981]

    P. Gács andL. Lovász, Khachiyan’s algorithm for linear programming,Math. Programming Studies 14 (1981) 61–68.

    MATH  Google Scholar 

  20. [1979]

    M. R. Garey andD. S. Johnson,Computers and intractability: a guide to the theory of NP-completeness, Freeman, San Francisco, 1979.

    Google Scholar 

  21. [1960]

    A. J. Hoffman, Some recent applications of the theory of linear inequalities to extremal combinatorial analysis, in:Combinatorial analysis, Proc. 10th Symp. on Appl. Math. Columbia Univ., 1958 (R. E. Bellman and M. Hall, Jr, eds.), Amer. Math. Soc., Providence, R. I., 1960, 113–127.

    Google Scholar 

  22. [1979]

    I. Holyer, The NP-completeness of edge-colouring,SIAM J. Comp., to appear.

  23. [1963]

    T. C. Hu, Multicommodity network flows,Operations Res. 11 (1963) 344–360.

    MATH  Article  Google Scholar 

  24. [1973]

    T. C. Hu, Two-commodity cut-packing problem,Discrete Math. 4 (1973) 108–109.

    MATH  Google Scholar 

  25. [1979]

    A. V. Karzanov, On the minimal number of arcs of a digraph meeting all its directed cutsets,to appear.

  26. [1979]

    L. G. Khachiyan, A polynomial algorithm in linear programming,Doklady Akademii Nauk SSSR 244 (1979) 1093–1096 (English translation:Soviet Math. Dokl. 20, 191–194).

    MATH  MathSciNet  Google Scholar 

  27. [1970c]

    E. L. Lawler, Optimal matroid intersections, in:Combinatorial structures and their applications, Proc. Intern. Conf. Calgary, Alb., 1969 (R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds.), Gordon and Breach, New York, 1970, 233–235.

    Google Scholar 

  28. [1976]

    E. L. Lawler,Combinatorial optimization: networks and matroids, Holt, Rinehart and Winston, New York, 1976.

    Google Scholar 

  29. [1972]

    L. Lovász, Normal hypergraphs and the perfect graph conjecture,Discrete Math. 2 (1972) 253–267.

    MATH  Article  MathSciNet  Google Scholar 

  30. [1975]

    L. Lovász, 2-Matchings and 2-covers of hypergraphs,Acta Math. Acad. Sci. Hungar. 26 (1975) 433–444.

    MATH  Article  MathSciNet  Google Scholar 

  31. [1978]

    L. Lovász, The matroid matching problem,Proc. Conf. Algebraic Methods in Graph Theory (Szeged, 1978), to appear.

  32. [1979]

    L. Lovász, On the Shannon capacity of a graph,IEEE Trans. on Information Theory 25 (1979) 1–7.

    MATH  Article  Google Scholar 

  33. [1981]

    L. Lovász, Perfect graphs, in:More selected topics in graph theory (L. W. Beineke and R. J. Wilson, eds), to appear

  34. [1976]

    C. L. Lucchesi, A minimax equality for directed graphs,Doctoral Thesis, Univ. Waterloo, Waterloo, Ont., 1976.

    Google Scholar 

  35. [1978]

    C. L. Lucchesi andD. H. Younger, A minimax relation for directed graphs,J. London Math. Soc. (2) 17 (1978) 369–374.

    MATH  Article  MathSciNet  Google Scholar 

  36. [1980]

    G. J. Minty, On maximal independent sets of vertices in claw-free graphs,J. Combinatorial Theory (B),28 (1980) 284–304.

    MATH  Article  MathSciNet  Google Scholar 

  37. [1954]

    T. S. Motzkin andI. J. Schoenberg, The relaxation method for linear inequalities,Canad. J. Math. 6 (1954) 393–404.

    MATH  MathSciNet  Google Scholar 

  38. [1979]

    H. Okamura andP. D. Seymour, Multicommodity flows in planar graphs,J. Combinatorial Theory (B), to appear.

  39. [1979]

    M. W. Padberg andM. R. Rao, Minimum cut-sets and b-matchings,to appear.

  40. [1980]

    A. Schrijver, A counterexample to a conjecture of Edmonds and Giles,Discrete Math. 32 (1980) 213–214.

    MATH  MathSciNet  Google Scholar 

  41. [1977]

    P. D. Seymour, The matroids with the max-flow min-cut property,J. Combinatorial Theory (B) 23 (1977) 189–222.

    MATH  Article  MathSciNet  Google Scholar 

  42. [1978]

    P. D. Seymour, A two-commodity cut theorem,Discrete Math. 23 (1978) 177–181.

    MATH  Article  MathSciNet  Google Scholar 

  43. [1970]

    N. Z. Shor, Convergence rate of the gradient descent method with dilatation of the space,Kibernetika 2 (1970) 80–85 (English translation:Cybernetics 6 (1970) 102–108).

    Google Scholar 

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Research by the third author was supported by the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.).

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Grötschel, M., Lovász, L. & Schrijver, A. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981). https://doi.org/10.1007/BF02579273

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AMS subject classification (1980)

  • 90 C XX, 05 C XX
  • 90 C 25, 90 C 10