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Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

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Abstract

We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal” solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0–1 optimization and to instances of the graph equipartition problem. The experiments show that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming-based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so.

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References

  1. Achterberg, T.: Constraint integer programming. PhD thesis, Technische Universität Berlin. http://opus.kobv.de/tuberlin/volltexte/2007/1611/ (2007)

  2. Achterberg T., Koch T., Martin A.: Branching rules revisited. Oper. Res. Lett. 33(1), 42–54 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Armbruster, M.: Branch-and-Cut for a semidefinite relaxation of the minimum bisection problem. PhD thesis, University of Technology Chemnitz (2007)

  4. Barahona F., Ladányi L.: Branch and cut based on the volume algorithm: Steiner trees in graphs and max-cut. RAIRO Oper. Res. 40(1), 53–73 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barahona F., Mahjoub A.R.: On the cut polytope. Math. Program. 36(2), 157–173 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barahona F., Jünger M., Reinelt G.: Experiments in quadratic 0–1 programming. Math. Program. 44(2, (Ser. A)), 127–137 (1989)

    Article  MATH  Google Scholar 

  7. Beasley J.E.: Or-library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)

    Article  Google Scholar 

  8. Beasley, J.E.: Or-library. http://people.brunel.ac.uk/~mastjjb/jeb/info.html (1990)

  9. Beasley, J.E.: Heuristic algorithms for the unconstrained binary quadratic programming problem. Technical report, The Management School, Imperial College, London SW7 2AZ, England (1998)

  10. Benson, S.J., Ye, Y., Zhang, X.: Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10(2), 443–461 (2000, electronic)

    Google Scholar 

  11. Billionnet A., Elloumi S.: Using a mixed integer quadratic programming solver for the unconstrained quadratic 0–1 problem. Math. Program. 109(1, Ser. A), 55–68 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Boros, E., Hammer, P.L., Tavares, G.: The pseudo-boolean optimization. http://rutcor.rutgers.edu/~pbo/ (2005)

  13. Boros E., Hammer P.L., Sun R., Tavares G.: A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO). Discrete Optim. 5(2), 501–529 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Burer, S., Monteiro, R.D., Zhang, Y.: Rank-two relaxation heuristics for max-cut and other binary quadratic programs. SIAM J. Optim. 12(2), 503–521 (2001/2002, electronic)

    Google Scholar 

  15. De Simone C., Diehl M., Jünger M., Mutzel P., Reinelt G., Rinaldi G.: Exact ground states of Ising spin glasses: new experimental results with a branch-and-cut algorithm. J. Stat. Phys. 80(1–2), 487–496 (1995)

    Article  MATH  Google Scholar 

  16. Delorme C., Poljak S.: Laplacian eigenvalues and the maximum cut problem. Math. Program. 62(3, Ser. A), 557–574 (1993)

    Article  MathSciNet  Google Scholar 

  17. Deza M.M., Laurent M.: Geometry of Cuts and Metrics. In: Algorithms and Combinatorics, vol. 15. Springer, Berlin (1997)

    Google Scholar 

  18. Elf, M., Gutwenger, C., Jünger, M., Rinaldi, G.: Branch-and-Cut Algorithms for Combinatorial Optimization and Their Implementation in ABACUS. In: Lecture Notes in Computer Science, vol. 2241, pp. 157–222. Springer, Heidelberg (2001)

  19. Fischer I., Gruber G., Rendl F., Sotirov R.: Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and equipartition. Math. Program. 105(2–3, Ser. B), 451–469 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Frangioni A., Lodi A., Rinaldi G.: New approaches for optimizing over the semimetric polytope. Math. Program. 104(2–3, Ser. B), 375–388 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Glover F., Kochenberger G., Alidaee B.: Adaptative memory tabu search for binary quadratic programs. Manage. Sci. 44(3), 336–345 (1998)

    Article  MATH  Google Scholar 

  22. Goemans, M.X., Williamson, D.P.: .878-approximation algorithms for max cut and max 2sat. In: Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pp. 422–431. Montreal, Quebec, Canada (1994)

  23. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995, preliminary version see [22]

    Google Scholar 

  24. Hansen P.: Methods of nonlinear 0–1 programming. Ann. Discrete Math. 5, 53–70 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  25. Helmberg, C.: Fixing variables in semidefinite relaxations. SIAM J. Matrix Anal. Appl. 21(3), 952–969 (2000, electronic)

    Google Scholar 

  26. Helmberg, C.: A cutting plane algorithm for large scale semidefinite relaxations. In: The Sharpest Cut, MPS/SIAM Ser. Optim., pp. 233–256. SIAM, Philadelphia, PA (2004)

  27. Helmberg C., Rendl F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Program. 82(3, Ser. A), 291–315 (1998)

    Article  MathSciNet  Google Scholar 

  28. Helmberg C., Rendl F., Vanderbei R.J., Wolkowicz H.: An interior-point method for semidefinite programming. SIAM J. Optim. 6(2), 342–361 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  29. Johnson D.S., Aragon C.R., McGeoch L.A., Schevon C.: Optimization by simulated annealing: an experimental evaluation. part i, graph partitioning. Oper. Res. 37(6), 865–892 (1989)

    Article  MATH  Google Scholar 

  30. Jünger, M., Reinelt, G., Rinaldi, G.: Lifting and separation procedures for the cut polytope. Technical report, Universität zu Köln (2006, in preparation)

  31. Karisch, S.E., Rendl, F.: Semidefinite programming and graph equipartition. In: Topics in semidefinite and interior-point methods (Toronto, ON, 1996). In: Fields Inst. Commun., vol. 18, pp. 77–95. American Mathematical Society, Providence (1998)

  32. Kim S., Kojima M.: Second order cone programming relaxation of nonconvex quadratic optimization problems. Optim. Methods Softw. 15(3-4), 201–224 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Laurent M.: The max-cut problem. In: Dell’Amico, M., Maffioli, F., Martello, S.(eds) Annotated Bibliographies in Combinatorial Optimization, pp. 241–259. Wiley, Chichester (1997)

    Google Scholar 

  34. Liers, F.: Contributions to determining exact ground-states of Ising spin-glasses and to their physics. PhD thesis, Universität zu Köln (2004)

  35. Liers F., Jünger M., Reinelt G., Rinaldi G.: Computing exact ground states of hard Ising spin glass problems by branch-and-cut. In: Hartmann, A., Rieger, H.(eds) New Optimization Algorithms in Physics, pp. 47–68. Wiley, London (2004)

    Chapter  Google Scholar 

  36. Muramatsu M., Suzuki T.: A new second-order cone programming relaxation for MAX-CUT problems. J. Oper. Res. Soc. Jpn 46(2), 164–177 (2003)

    MATH  MathSciNet  Google Scholar 

  37. Pardalos P.M., Rodgers G.P.: Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45(2), 131–144 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  38. Pardalos P.M., Rodgers G.P.: Parallel branch and bound algorithms for quadratic zero-one programs on the hypercube architecture. Ann. Oper. Res. 22(1–4), 271–292 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  39. Poljak S., Rendl F.: Solving the max-cut problem using eigenvalues. Discrete Appl. Math. 62(1–3), 249–278 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  40. Poljak S., Rendl F.: Nonpolyhedral relaxations of graph-bisection problems. SIAM J. Optim. 5(3), 467–487 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  41. Rendl F.: Semidefinite programming and combinatorial optimization. Appl. Numer. Math. 29(3), 255–281 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  42. Rendl, F., Rinaldi, G., Wiegele, A.: A branch and bound algorithm for Max-Cut based on combining semidefinite and polyhedral relaxations. In: Integer programming and combinatorial optimization. Lecture Notes in Computer Science, vol. 4513, pp. 295–309. Springer, Berlin (2007)

  43. Rinaldi, G.: Rudy. http://www-user.tu-chemnitz.de/~helmberg/rudy.tar.gz (1998)

  44. Wiegele, A.: Nonlinear optimization techniques applied to combinatorial optimization problems. PhD thesis, Alpen-Adria-Universität Klagenfurt (2006)

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Authors and Affiliations

  1. Alpen-Adria-Universität Klagenfurt, Institut für Mathematik, Universitätsstr. 65-67, 9020, Klagenfurt, Austria

    Franz Rendl & Angelika Wiegele

  2. Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti” del CNR, viale Manzoni 30, 00185, Rome, Italy

    Giovanni Rinaldi

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  1. Franz Rendl
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  2. Giovanni Rinaldi
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  3. Angelika Wiegele
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Correspondence to Angelika Wiegele.

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Supported in part by the EU project Algorithmic Discrete Optimization (ADONET), MRTN-CT-2003-504438.

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Rendl, F., Rinaldi, G. & Wiegele, A. Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121, 307–335 (2010). https://doi.org/10.1007/s10107-008-0235-8

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