Advertisement

Annals of Operations Research

, Volume 33, Issue 3, pp 151–180 | Cite as

The max-cut problem and quadratic 0–1 optimization; polyhedral aspects, relaxations and bounds

  • Endre Boros
  • Peter L. Hammer
Section III Graph-Theoretical Aspects Of TND

Abstract

Given a graphG, themaximum cut problem consists of finding the subsetS of vertices such that the number of edges having exactly one endpoint inS is as large as possible. In the weighted version of this problem there are given real weights on the edges ofG, and the objective is to maximize the sum of the weights of the edges having exactly one endpoint in the subsetS. In this paper, we consider the maximum cut problem and some related problems, likemaximum-2-satisfiability, weighted signed graph balancing. We describe the relation of these problems to the unconstrained quadratic 0–1 programming problem, and we survey the known methods for lower and upper bounds to this optimization problem. We also give the relation between the related polyhedra, and we describe some of the known and some new classes of facets for them.

Keywords

Endpoint Programming Problem Related Problem Weighted Version Signed Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    V.D. Agrawal and S.C. Seth,Test Generation for VLSI Chips (IEEE Computer Society Press, 1988).Google Scholar
  2. [2]
    N. Alon, The CW-inequalities for vectors inl 1, Eur. J. Combinatorics 10(1990).Google Scholar
  3. [3]
    W.P. Adams and P.M. Dearing, On the equivalence between roof duality and Lagrangian duality for unconstrained 0–1 quadratic programming problems, Technical Report No. URI-061, The Clemson University (1988).Google Scholar
  4. [4]
    E. Balas and J.B. Mazzola, Nonlinear 0–1 programming: I. Linearization techniques and II. Dominance relations and algorithms, Math. Progr. 30(1984)1–45.Google Scholar
  5. [5]
    M. Balinski, On a selection problem, Manag. Sci. 17(1970)230–231.Google Scholar
  6. [6]
    F. Barahona, The max-cut problem in graphs not contractible toK 5, Oper. Res. Lett. 2(1983)107–111.Google Scholar
  7. [7]
    F. Barahona, A solvable case of quadratic 0–1 programming, Discr. Appl. Math. 13(1986)23–26.Google Scholar
  8. [8]
    F. Barahona, On cuts and matchings in planar graphs, Research Report No. 88503-OR, Institut für Operations Research, Universität Bonn (1988).Google Scholar
  9. [9]
    F. Barahona, M. Grötschel, M. Jünger and G. Reinelt, An application of combinatorial optimization to statistical physics and circuit layout design, Oper. Res. 36(1988)493–513.Google Scholar
  10. [10]
    F. Barahona, M. Grötschel and A.R. Mahjoub, Facets of the bipartite subgraph polytope, Math. Oper. Res. 10(1985)340–358.Google Scholar
  11. [11]
    F. Barahona, M. Jüngert and G. Reinelt, Experiments in quadratic 0–1 programming, Res. Rep. No. 7, Institut für Mathematik, Universität Augsburg (1987).Google Scholar
  12. [12]
    F. Barahona and A.R. Mahjoub, On the cut polytope, Math. Progr. 36(1986)157–173.Google Scholar
  13. [13]
    A. Billionnet and B. Jaumard, A decomposition method for minimizing quadratic pseudo-Boolean functions, Oper. Res. Lett. 8(1989)161–163.Google Scholar
  14. [14]
    S.H. Bokhari,Assignment Problems in Parallel and Distributed Computing (Kluwer Academic, Boston/Dordrecht/Lancaster, 1987).Google Scholar
  15. [15]
    E. Boros, Y. Crama and P.L. Hammer, Upper bounds for quadratic 0–1 maximization problems, Oper. Res. Lett. 9(1990)73–79.Google Scholar
  16. [16]
    E. Boros, Y. Crama and P.L. Hammer, Chvátal cuts, and odd cycle inequalities in quadratic 0–1 optimization, SIAM J. Discr. Math., to appear.Google Scholar
  17. [17]
    E. Boros and P.L. Hammer, On clustering problems with connected optima in Euclidean spaces, Discr. Math. 75(1989)81–88.Google Scholar
  18. [18]
    E. Boros and P.L. Hammer, A max-flow approach to improved roof-duality in quadratic 0–1 minimization, RUTCOR Research Report 15-89, Rutgers University, New Brunswick, NJ (1989).Google Scholar
  19. [19]
    E. Boros and P.L. Hammer, Cut-polytopes, Boolean quadric polytopes and nonnegative quadratic pseudo-Boolean functions, Math. Oper. Res., to appear.Google Scholar
  20. [20]
    E. Boros and P.L. Hammer, On extremal nonnegative quadratic pseudo-Boolean functions, Manuscript (1990).Google Scholar
  21. [21]
    E. Boros, P.L. Hammer and X. Sun, The DDT method for quadratic 0–1 minimization, RUTCOR Research Report 39-89, Rutgers University, New Brunswick, NJ (1989).Google Scholar
  22. [22]
    J.-M. Bourjolly, Integral and fractional node-packings, and pseudo-Boolean programs, Ph.D. Dissertation, Waterloo, Ontario (1986).Google Scholar
  23. [23]
    J.-M. Bourjolly, P.L. Hammer, W.R. Pulleyblank and B. Simeone, Combinatorial methods for bounding a quadratic pseudo-Boolean function, Research Report CORR 89-21, University of Waterloo (1989).Google Scholar
  24. [24]
    S.T. Chakradhar, M.L. Bushnell and V.D. Agrawal, Automatic test pattern generation using neural networks, in:Proc. Int. Conf. on Computer-Aided Design (IEEE, 1988), pp. 416–419.Google Scholar
  25. [25]
    C.K. Cheng, S.Z. Yao and T.C. Hu, The orientation of modules based on graph decomposition, IEEE Trans. Comput. C-40(1991)774–780.Google Scholar
  26. [26]
    S. Chopra and M.R. Rao, Facets of thek-partition polytope (June, 1989).Google Scholar
  27. [27]
    V. Chvátal, W. Cook and M. Hartmann, On cutting-plane proofs in combinatorial optimization, Linear Algebra Appl. 114/115(1989)455–499.Google Scholar
  28. [28]
    M. Conforti, M.R. Rao and A. Sassano, The equipartition polytope: Part I and II, R.194–195, IASICNR, Roma, Italy (1987).Google Scholar
  29. [29]
    Y. Crama, P. Hansen and B. Jaumard, The basic algorithm for pseudo-Boolean programming revisited, RUTCOR Research Report 54-88, Rutgers University, New Brunswick, NJ (1988).Google Scholar
  30. [30]
    P.M. Dearing, P.L. Hammer and B. Simeone, Boolean and graph-theoretic formulations of the simple plant location problem, RUTCOR Research Report 3-88, Rutgers University, New Brunswick, NJ (1988).Google Scholar
  31. [31]
    J.S. DeCani, Maximum likelihood paired comparison ranking by linear programming, Biometrika 56(1969)537–545.Google Scholar
  32. [32]
    C. De Simone, The cut polytope and the Boolean quadric polytope, Discr. Math. 79(1990)71–75.Google Scholar
  33. [33]
    M. Deza and M. Laurent, Facets for the complete cut cone I, Research Memorandum RMI 88-13, Department of Mathematical Engineering, University of Tokyo (1988).Google Scholar
  34. [34]
    M. Deza and M. Laurent, Facets for the complete cut cone II: Clique-webb inequalities, Document du Lamsade n. 57, Université Paris Dauphine (1989).Google Scholar
  35. [35]
    M. Deza and M. Laurent, New results on facets of the cut cone, Report No. B-227, Department of Information Sciences, Tokyo Institute of Technology (1989).Google Scholar
  36. [36]
    M. Deza, C. De Simone and M. Laurent, Collapsing and lifting for the cut cone, IASI Report No. 265, Rome (1989).Google Scholar
  37. [37]
    M. Deza, M. Grötschel and M. Laurent, Clique-webb facets for multicut polytopes, Research Report No. 186, Universität Augsburg (1989).Google Scholar
  38. [38]
    C. Ebenegger, P.L. Hammer and D. de Werra, Pseudo-Boolean functions and stability of graphs, Ann. Discr. Math. 19(1984)83–98.Google Scholar
  39. [39]
    R.W. Floyd and L. Steinberg, An adaptive algorithm for spatial greyscale, Proc. SID 17(1976)75–77.Google Scholar
  40. [40]
    J. Fonlupt, A.R. Mahjoub and J.-P. Uhry, Composition of graphs and the bipartite subgraph polytope, Research Report No. 459, Lab. ARTEMIS (IMAG), Univ. de Grenoble (1984).Google Scholar
  41. [41]
    R. Fortet, L'algèbre de Boole et ses applications en recherche operationelle, Cahiers du Centre d'Etudes de Recherche Operationelle 1(1959)5–36.Google Scholar
  42. [42]
    A.M.H. Gerards, Testing the odd bicycle wheel inequality for the bipartite subgraph polytope, Math. Oper. Res. 10(1985)359–360.Google Scholar
  43. [43]
    M. Grötschel, L. Lovász and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1(1981)169–197.Google Scholar
  44. [44]
    M. Grötschel and W.R. Pulleyblank, Weakly bipartite graphs and the max-cut problem, Oper. Res. Lett. 1(1981)23–27.Google Scholar
  45. [45]
    M. Grötschel and Y. Wakabayashi, Facets of the clique-partitioning polytope, Report No. 6, Institute für Mathematik, Universität Augsburg (1987).Google Scholar
  46. [46]
    F. Hadlock, Finding a maximum cut of a planar graph in polynomial time, SIAM J. Comput. 4(1975)221–225.Google Scholar
  47. [47]
    P.L. Hammer, Some network flow problems solved with pseudo-Boolean programming, Oper. Res. 13(1965)388–399.Google Scholar
  48. [48]
    P.L. Hammer, Plant location — A pseudo-Boolean approach, Israel J. Tech. 6(1968)330–332.Google Scholar
  49. [49]
    P.L. Hammer, Pseudo-Boolean remarks on balanced graphs, Int. Series Num. Math. 36(1977)69–78.Google Scholar
  50. [50]
    P.L. Hammer and P. Hansen, Logical relations in quadratic 0–1 programming, Rev. Roum. Math. Pures Appl. 26(1981)421–429.Google Scholar
  51. [51]
    P.L. Hammer, P. Hansen and B. Simeone, On vertices belonging to all or to no maximum stable sets of a graph, SIAM J. Alg. Discr. Meth. 3(1982)511–522.Google Scholar
  52. [52]
    P.L. Hammer, P. Hansen and B. Simeone, Roof duality, complementation and persistency in quadratic 0–1 optimization, Math. Progr. 28(1984)121–155.Google Scholar
  53. [53]
    P.L. Hammer and B. Kalantari, A bound on the roof duality gap, RUTCOR Research Report 46-87, Rutgers University. New Brunswick, NJ (1987).Google Scholar
  54. [54]
    P.L. Hammer, I. Rosenberg and S. Rudeanu, On the determination of the minima of pseudo-Boolean functions, Stud. Cerc. Mat. 14(1963)359–364, in Romanian.Google Scholar
  55. [55]
    P.L. Hammer, I. Rosenberg and S. Rudeanu, Application of discrete linear programming to the minimization of Boolean functions, Rev. Roum. Math. Pures Appl. 8(1963)459–475, in Russian.Google Scholar
  56. [56]
    P.L. Hammer and B. Simeone, Quadratic functions on binary variables, RUTCOR Research Report 20-87, Rutgers University, New Brunswick, NJ (1987).Google Scholar
  57. [57]
    P. Hansen, Methods of nonlinear 0–1 programming, Ann. Discr. Math. 5(1979)53–70.Google Scholar
  58. [58]
    P. Hansen, S.H. Lu and B. Simeone, On the equivalence of paved-duality and standard linearization in nonlinear optimization, Discr. Appl. Math., to appear.Google Scholar
  59. [59]
    F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2(1953/54)143–146.Google Scholar
  60. [60]
    J.J. Hopfield, Artificial neural networks, IEEE Circuits Dev. Mag. 4(1988)3–10.Google Scholar
  61. [61]
    O.H. Ibarra and S.K. Sahni, Polynomially complete fault detection problems, IEEE Trans. Comput. C-24(1975)242–249.Google Scholar
  62. [62]
    J.F. Jarvis, C.N. Júdice and W.H. Ninke, A survey of techniques for the display of continuous tone pictures on bilevel displays, Comput. Graphics Image Proc. 5(1976)13–40.Google Scholar
  63. [63]
    R.M. Karp, Reducibility among combinatorial problems,Complexity of Computer Computations, ed. R.E. Miller and J.W. Thatcher (Plenum, New York, 1972), pp. 85–103.Google Scholar
  64. [64]
    A.V. Karzanov, Metrics and undirected cuts, Math. Progr. 32(1985)183–198.Google Scholar
  65. [65]
    D.J. Laughhunn, Quadratic binary programming with applications to capital budgeting problems, Oper. Res. 18(1970)454–461.Google Scholar
  66. [66]
    S.H. Lu and A.C. Williams, Roof duality for 0–1 nonlinear optimization, Math. Progr. 37(1987)357–360.Google Scholar
  67. [67]
    S.H. Lu and A.C. Williams, Roof duality and linear relaxation for quadratic and polynomial 0–1 optimization, RUTCOR Research Report 8-87, Rutgers University, New Brunswick, NJ (1987).Google Scholar
  68. [68]
    G.I. Orlova and Y.G. Dorfman, Finding the maximum cut in a graph, Izv. Akad. Nauk SSSR 72(1975)155–159, in Russian [English transl.: Eng. Cybernetics 10(1975)502–506].Google Scholar
  69. [69]
    M. Padberg, The Boolean quadric polytope: Some characteristics, facets and relatives, Math. Progr. 45(1989)139–171.Google Scholar
  70. [70]
    S.G. Papaioannou, Optimal test generation in combinatorial networks by pseudo-Boolean programming, IEEE Trans. Comput. C-26(1977)553–560.Google Scholar
  71. [71]
    L. Personnaz, I. Guyon and G. Dreyfus, Collective computational properties of neural networks: New learning mechanisms, Phys. Rev. A34(1986)4217–4228.Google Scholar
  72. [72]
    J.C. Picard and H.D. Ratliff, A cut approach to the rectilinear facility location problem, Oper. Res. 26(1978)422–433.Google Scholar
  73. [73]
    J. Rhys, A selection problem of shared fixed costs and networks, Manag. Sci. 17(1970)200–207.Google Scholar
  74. [74]
    P.D. Seymour, Matroids and multicommodity flows, Eur. J. Combinatorics 2(1981)257–290.Google Scholar
  75. [75]
    B. Simeone, Consistency of quadratic Boolean equations and the König-Egerváry property for graphs, Ann. Discr. Math. 25(1985)281–290.Google Scholar
  76. [76]
    S.R. Sreinberg, Biomedical image processing, IEEE Computer 15(1982)47–56.Google Scholar
  77. [77]
    H.S. Stone, Multiprocessing scheduling with the aid of network flow algorithms, IEEE Trans. Software Eng. SE-3(1977)85–93.Google Scholar
  78. [78]
    H.S. Stone, Program assignment in three-processor systems and tricutset partitioning of graphs, Technical Report No. ECE-CS-77-7, Department of Electrical and Computer Engineering, University of Massachusetts, Amherst (1977).Google Scholar
  79. [79]
    H.S. Stone, Critical load factors in two-processor distributed systems, IEEE Trans. Software Eng. SE-4(1978)254–258.Google Scholar
  80. [80]
    A. Warszawski, Pseudo-Boolean solutions to multidimensional location problems, Oper. Res. 22(1974)1081–1085.Google Scholar
  81. [81]
    A.C. Williams, Quadratic 0–1 programming using the roof dual with computational results, RUTCOR Research Report 8-85, Rutgers University, New Brunswick, NJ (1985).Google Scholar

Copyright information

© J.C. Baltzer AG, Scientific Publishing Company 1991

Authors and Affiliations

  • Endre Boros
    • 1
  • Peter L. Hammer
    • 2
  1. 1.DIMACS and RUTCORRutgers UniversityNew BrunswickUSA
  2. 2.RUTCOR, Rutgers UniversityNew BrunswickUSA

Personalised recommendations