A semidefinite programming based polyhedral cut and price approach for the Maxcut problem

Computational Optimization and Applications

Volumn 33, Issue 1, 2006, Pages 51-71

  Krishnan, Kartik ^a   Mitchell, John E ^b  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

^b RENSSELAER POLYTECHNIC INSTITUTE   (United States)

Author keywords

Column generation; Combinatorial optimization; Cutting plane methods; Semidefinite programming

Indexed keywords

COLUMN GENERATION; COMBINATORIAL OPTIMIZATION; CUTTING PLANE METHODS; SEMIDEFINITE PROGRAMMING;

ALGORITHMS; COMPUTATIONAL COMPLEXITY; LINEAR PROGRAMMING; OPTIMIZATION; PROBLEM SOLVING; RELAXATION PROCESSES;

MATHEMATICAL PROGRAMMING;

EID: 32544446882     PISSN: 09266003     EISSN: 15732894     Source Type: Journal    
DOI: 10.1007/s10589-005-5958-3     Document Type: Conference Paper

References (45)

1. M.F. Anjos and H. Wolkowicz, "Strengthened semidefinite relaxations via a second lifting for the max-cut problem," Discrete Applied Mathematics, vol. 119, pp. 79-106, 2002.
2. D. Avis and J. Umemoto, "Stronger linear programming relaxations of max-cut," Mathematical Programming, vol. 97, pp. 451-469, 2003.
3. F. Barahona, M. Grötschel, M. Jünger, and G. Reinelt, "An application of combinatorial optimization to statistical physics and circuit layout design," Operations Research, vol. 36, pp. 493-513, 1998.
4. F. Barahona and A. R. Mahjoub, "On the cut polytope," Mathematical Programming, vol. 36, pp. 157-173, 1986.
5. J. Berry and M. Goldberg, "Path optimization for graph partitioning problems," Discrete Applied Mathematics, vol. 90, pp. 27-50, 1999.
6. B. Borchers, "SDPLIB 1.2, a library of semidefinite programming test problems," Optimization Methods and Software, vol. 11, pp. 683-690, 1999.
7. K. C. Chang and D. Z. Du, "Efficient algorithms for layout assignment problems," IEEE Transactions on Computer Aided Design, vol. 6, pp. 67-78, 1987.
8. CPLEX Optimization Inc., "CPLEX Linear Optimizer and Mixed Integer Optimizer," Suite 279, 930 Tahoe Blvd. Bldg 802, Incline Village, NV 89541.
9. C. De Simone, M. Diehl, M. Jünger, P. Mutzel, G. Reinelt, and G. Rinaldi, "Exact ground states of Ising spin glasses: New experimental results with a branch and cut algorithm," Journal of Statistical Physics, vol. 80, pp. 487-496, 1995.
10. C. De Simone, M. Diehl, M. Jünger, P. Mutzel, G. Reinelt, and G. Rinaldi, "Exact ground states of two-dimensional ± J Ising spin glasses," Journal of Statistical Physics, vol. 84, pp. 1363-1371, 1996.
11. M.M. Deza and M. Laurent, Geometry of Cuts and Metrics, Springer Verlag: Berlin Heidelberg, 1997.
12. S. Elhedhli and J.L. Goffin, "The integration of an interior-point cutting plane method within a branch-and-price algorithm," Mathematical Programming, vol. 100, pp. 267-294, 2004.
13. M.X. Goemans and D.P. Williamson, "Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming," J. Assoc. Comput. Mach., vol. 42, pp. 1115-1145, 1995.
14. B. Grone, C.R. Johnson, E. Marques de Sa, and H. Wolkowicz, "Positive definite completions of partial Hermitian matrices," Linear Algebra and its Applications, vol. 58, pp. 109-124, 1984.
15. M. Grötschel, L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag: Berlin, Germany, 1988.
16. G. Gruber, "On semidefinite programming and applications in combinatorial optimization," PhD thesis, University of Technology, Graz, Austria, April 2000.
17. J. Hrâstad, "Some optimal inapproximability results," Journal of the ACM, vol. 48, no. 4, pp. 798-859, 2001.
18. C. Helmberg, "Semidefinite programming for combinatorial optimization," Technical Report ZR-00-34, TU Berlin, Konrad-Zuse-Zentrum, Berlin, October 2000. Habilitationsschrift. Available from http://www.zib.de/ PaperWeb/abstracts/ZR-00-34.
19. C. Helmberg, "A cutting plane algorithm for large scale semidefinite relaxations," In The Sharpest Cut, edited by M. Grötschel, Festschrift in honor of M. Padberg's 60th birthday, MPS-SIAM 2004, pp. 233-256.
20. C. Helmberg and F. Rendl, "Solving quadratic (0,1)-problems by semidefinite programs and cutting planes," Mathematical Programming, vol. 82, pp. 291-315, 1998.
21. H. Karloff, "How good is the Goemans-Williamson MAX CUT algorithm?" SIAM Journal on Computing, vol. 29, pp. 336-350, 1999.
22. R.M. Karp, "Reducibility among combinatorial problems," In: Complexity of Computer Computations, edited by R.E. Miller and J.W. Thatcher, Plenum Press; New York, 1972, pp. 85-103.
23. J.E. Kelley, "The cutting plane method for solving convex programs," Journal of SIAM, vol. 8, pp. 703-712, 1960.
24. B.W. Kernighan and S. Lin, "An efficient heuristic procedure for partitioning graphs," Bell System Technical Journal, vol. 49, pp. 291-307, 1970.
25. K. Krishnan, "Linear programming approaches to semidefinite programming problems," PhD thesis, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, July 2002.
26. K. Krishnan and J.E. Mitchell, "Semi-infinite linear programming approaches to semidefinite programming (SDP) problems," Fields Institute Communications Series, Volume 37, Novel approaches to hard discrete optimization problems, edited by P. Pardalos and H. Wolkowicz, pp. 121-140, 2003.
27. K. Krishnan and J.E. Mitchell, "A unifying framework for several cutting plane methods for semidefinite programming". Optimization Methods and Software, vol. 21, pp. 57-74, 2006.
28. K. Krishnan and J.E. Mitchell, "Properties of a cutting plane method for semidefinite programming," Technical report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, May 2003. Available from http://www4.ncsu.edu/~kksivara/.
29. K. Krishnan and T. Terlaky, "Interior point and semidefinite approaches in combinatorial optimization," GERAD 25th anniversary volume on Graph Theory and Combinatorial Optimization, edited by D. Avis, A. Hertz, and O. Marcotte, Springer, pp. 101-157, 2005.
30. J.B. Lasserre, "Global optimization with polynomials and the problem of moments," SIAM Journal on Optimization, vol. 11 pp. 796-817, 2001.
31. J.B. Lasserre, "An explicit equivalent semidefinite program for nonlinear 0-1 programs," SIAM Journal on Optimization, vol. 12, pp. 756-769, 2002.
32. M. Laurent, "A journey from some classical results to the Goemans Williamson approximation algorithm for maxcut," Lecture given at CORE, Louvain La Neuve, September 1-12 2003. Available from http://homepages.cwl.nl/ ~monique/
33. M. Laurent and S. Poljak, "On a positive semidefinite relaxation of the cut polytope," Linear Algebra and its Applications, vol. 223, pp. 439-461, 1995.
34. C. Lemarechal, "Non-differentiable Optimization" in edited by Optimization, G.L. Nemhauser, A.H.G. Rinnooy Kan, and M.J. Todd, eds., North-Holland, New York, 1989 pp. 529-572.
35. J.E. Mitchell, "Computational experience with an interior point cutting plane algorithm," SIAM Journal on Optimization, vol. 10, pp. 1212-1227, 2000.
36. Y.E. Nesterov, "Semidefinite relaxation and nonconvex quadratic optimization," Optimization Methods and Software, vol. 9, pp. 141-160, 1998.
37. C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Dover Publications Inc.: Mineola, New York, 1998.
38. G. Pataki, "On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues," Mathematics of Operations Research, vol. 23, pp. 339-358, 1998.
39. G. Pataki and S.H. Schmieta, "The DIMACS library of mixed semidefinite, quadratic, and linear programs," Available at http://dimacs.ratgers.edu/Challenges/Seventh/Instances/
40. R.Y. Pinter, "Optimal layer assignment for interconnect," Journal of VLSI Computational Systems, vol. 1, pp. 123-137, 1984.
41. S. Poljak and Z. Tuza, "Maximum cuts and large bipartite subgraphs," In "Combinatorial Optimization", volume 20 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, AMS/DIMACS, pp. 181-244, 1995.
42. S. Poljak and Z. Tuza, "The expected relative error of the polyhedral approximation of the maxcut problem," Operations Research Letters, vol. 16, pp. 191-198, 1994.
43. K.C. Toh, M. J. Todd, and R. Tutuncu, "SDPT3- a Matlab software package for semidefinite programming," Optimization Methods and Software, vol. 11, pp. 545-581, 1999.
44. L. Wolsey, "Integer Programming," John Wiley & Sons, Inc.: New York, 1998.
45. Y. Zhang, "User's guide to LIPSOL: Linear programming interior point solvers v0.4," Optimization Methods and Software, vol. 11, pp. 385-396, 1999.