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

Mathematical Programming

Volumn 121, Issue 2, 2010, Pages 307-335

  Rendl, Franz ^a   Rinaldi, Giovanni ^b   Wiegele, Angelika ^a  

^a UNIVERSITY OF KLAGENFURT   (Austria)

^b CNR   (Italy)

Author keywords

Cut polytope; Maximum cut; Semidefinite programming; Unconstrained binary quadratic optimization

Indexed keywords

EXPERIMENTS; LINEAR PROGRAMMING; NUMERICAL METHODS; OPTIMIZATION;

BOUNDING PROCEDURES; CUT POLYTOPE; LAGRANGIAN DUALITY; MAXIMUM CUTS; SEMI-DEFINITE PROGRAMMING; SEMIDEFINITE RELAXATION; TRIANGLE INEQUALITY; UNCONSTRAINED BINARY QUADRATIC OPTIMIZATIONS;

PROBLEM SOLVING;

EID: 77954940445     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-008-0235-8     Document Type: Article

References (44)

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)
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)
5. Barahona, F., Mahjoub, A.R.: On the cut polytope. Math. Program. 36(2), 157-173 (1986)
6. Barahona, F., Jünger, M., Reinelt, G.: Experiments in quadratic 0-1 programming. Math. Program. 44(2, (Ser. A)), 127-137 (1989)
7. Beasley, J.E.:Or-library: distributing test problems by electronicmail. J. Oper. Res. Soc. 41(11), 1069-1072 (1990)
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)
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)
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)
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)
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)
16. Delorme, C., Poljak, S.: Laplacian eigenvalues and the maximum cut problem. Math. Program. 62(3, Ser. A), 557-574 (1993)
17. Deza, M.M., Laurent, M.: Geometry of Cuts and Metrics. In: Algorithms and Combinatorics, vol. 15. Springer, Berlin (1997)
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)
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)
21. Glover, F., Kochenberger, G., Alidaee, B.: Adaptative memory tabu search for binary quadratic programs. Manage. Sci. 44(3), 336-345 (1998)
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])
24. Hansen, P.: Methods of nonlinear 0-1 programming. Ann. Discrete Math. 5, 53-70 (1979)
25. Helmberg, C.: Fixing variables in semidefinite relaxations. SIAM J.Matrix Anal. Appl. 21(3), 952-969 (2000, electronic)
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)
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)
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)
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)
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)
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)
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)
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)
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)
39. Poljak, S., Rendl, F.: Solving the max-cut problem using eigenvalues. Discrete Appl. Math. 62(1-3), 249-278 (1995)
40. Poljak, S., Rendl, F.: Nonpolyhedral relaxations of graph-bisection problems. SIAM J. Optim. 5(3), 467-487 (1995)
41. Rendl, F.: Semidefinite programming and combinatorial optimization. Appl.Numer.Math. 29(3), 255-281 (1999)
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)