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Exploiting sparsity for the min k-partition problem

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Abstract

The minimum k-partition problem is a challenging combinatorial problem with a diverse set of applications ranging from telecommunications to sports scheduling. It generalizes the max-cut problem and has been extensively studied since the late sixties. Strong integer formulations proposed in the literature suffer from a large number of constraints and variables. In this work, we introduce two more compact integer linear and semidefinite reformulations that exploit the sparsity of the underlying graph and develop theoretical results leveraging the power of chordal decomposition. Numerical experiments show that the new formulations improve upon state-of-the-art.

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References

  1. Anjos, M.F., Ghaddar, B., Hupp, L., Liers, F., Wiegele, A.: Solving \(k\)-way graph partitioning problems to optimality: the impact of semidefinite relaxations and the bundle method. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization: Festschrift for Martin Grötschel, pp. 355–386. Springer, Heidelberg (2013)

    Chapter  MATH  Google Scholar 

  2. Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math. 23(1), 11–24 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Ben-Ameur, W., Ouorou, A., Wang, G.: Convex and concave envelopes: revisited and new perspectives. Oper. Res. Lett. 45(5), 421–426 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berge, C.: Some classes of perfect graphs. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp. 155–165. Academic Press, New York (1967)

    Google Scholar 

  6. Bienstock, D., Muñoz, G.: LP formulations for polynomial optimization problems. SIAM J. Optim. 28(2), 1121–1150 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bienstock, D., Ozbay, N.: Tree-width and the Sherali–Adams operator. Discrete Optim. 1(1), 13–21 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Blair, J.R., Peyton, B.: An introduction to chordal graphs and clique trees. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds.) Graph Theory and Sparse Matrix Computation, pp. 1–29. Springer, New York (1993)

    Google Scholar 

  9. Bodlaender, H.L., Koster, A.M.: Treewidth computations I. Upper bounds. Inf. Comput. 208(3), 259–275 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carlson, R., Nemhauser, G.L.: Scheduling to minimize interaction cost. Oper. Res. 14(1), 52–58 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chopra, S., Rao, M.: Facets of the \(k\)-partition polytope. Discrete Appl. Math. 61(1), 27–48 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chopra, S., Rao, M.R.: The partition problem. Math. Program. 59(1–3), 87–115 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cplex: Cplex 12.7 user’s manual. ILOG (2010)

  14. Deza, M., Grötschel, M., Laurent, M.: Complete descriptions of small multicut polytopes. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics, pp. 221–252. AMS, Providence (1991)

    Google Scholar 

  15. Deza, M., Grötschel, M., Laurent, M.: Clique-web facets for multicut polytopes. Math. Oper. Res. 17(4), 981–1000 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deza, M., Laurent, M.: Geometry of Cuts and Metric Embeddings. Springer, New York (1997)

    Book  MATH  Google Scholar 

  17. Eisenblätter, A.: The semidefinite relaxation of the \(k\)-partition polytope is strong. In: Cook, W., Schulz, A. (eds.) Proceedings of the 9th International Conference on Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 2337, pp. 273–290. Springer, Heidelberg (2002)

  18. Fairbrother, J., Letchford, A.N.: Projection results for the \(k\)-partition problem. Discrete Optim. 26, 97–111 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Frieze, A., Jerrum, M.: Improved approximation algorithms for max-\(k\)-cut and max bisection. Algorithmica 18(1), 67–81 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion I: general framework. SIAM J. Optim. 11(3), 647–674 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fulkerson, D., Gross, O.: Incidence matrices and interval graphs. Pac. J. Math. 15(3), 835–855 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1(2), 180–187 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ghaddar, B., Anjos, M.F., Liers, F.: A branch-and-cut algorithm based on semidefinite programming for the minimum \(k\)-partition problem. Ann. Oper. Res. 188(1), 155–174 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)

    MATH  Google Scholar 

  26. Grone, R., Johnson, C.R., Sá, E.M., Wolkowicz, H.: Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58, 109–124 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hijazi, H., Wang, G., Coffrin, C.: Gravity: A mathematical modeling language for optimization and machine learning. Machine Learning Open Source Software Workshop at NIPS 2018 (2018)

  28. Kaibel, V., Peinhardt, M., Pfetsch, M.E.: Orbitopal fixing. Discrete Optim. 8(4), 595–610 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kim, S., Kojima, M., Mevissen, M., Yamashita, M.: Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion. Math. Program. 129(1), 33–68 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Koster, A.M., Bodlaender, H.L., Van Hoesel, S.P.: Treewidth: computational experiments. Electron. Notes Discrete Math. 8, 54–57 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lasserre, J.B.: Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Laurent, M.: A connection between positive semidefinite and euclidean distance matrix completion problems. Linear Algebra Appl. 273(1), 9–22 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Michael, R.G., David, S.J.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  34. MOSEK: MOSEK Fusion API for C++. http://docs.mosek.com (2017)

  35. Nakata, K., Fujisawa, K., Fukuda, M., Kojima, M., Murota, K.: Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results. Math. Program. 95(2), 303–327 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of computing, pp. 229–234. ACM, New York (1988)

  37. Rinaldi, G.: Rudy, a graph generator. https://www-user.tu-chemnitz.de/helmberg/sdp_software.html (1998)

  38. de Sousa, V.J.R., Anjos, M.F., Le Digabel, S.: Computational study of valid inequalities for the maximum \(k\)-cut problem. Ann. Oper. Res. 265(1), 5–27 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Vandenberghe, L., Andersen, M.S., et al.: Chordal graphs and semidefinite optimization. Found. Trends Optim. 1(4), 241–433 (2015)

    Article  Google Scholar 

  40. Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17(1), 218–242 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, G.: Relaxations in mixed-integer quadratically constrained programming and robust programming. Ph.D. thesis, Evry, Institut national des télécommunications (2016)

  42. Wang, G., Ben-Ameur, W., Ouorou, A.: A Lagrange decomposition based branch and bound algorithm for the optimal mapping of cloud virtual machines. Eur. J. Oper. Res. 276(1), 28–39 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang, G., Hijazi, H.: An implementation for solving the min \(k\)-partition problem. https://zenodo.org/record/3252476. https://doi.org/10.5281/zenodo.3252476 (2019)

  44. Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Algebr. Discrete Methods 2(1), 77–79 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the associate editor and the anonymous referees for their helpful suggestions that greatly improved the quality of the paper. This research was partly funded by the Australia Indonesia Centre.

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Authors and Affiliations

  1. The Australian National University, Acton, Canberra, 2601, Australia

    Guanglei Wang & Hassan Hijazi

  2. Los Alamos National Laboratory, Los Alamos, NM, USA

    Hassan Hijazi

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  1. Guanglei Wang
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  2. Hassan Hijazi
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Correspondence to Guanglei Wang.

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Wang, G., Hijazi, H. Exploiting sparsity for the min k-partition problem. Math. Prog. Comp. 12, 109–130 (2020). https://doi.org/10.1007/s12532-019-00165-3

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