Computational study of valid inequalities for the maximum k-cut problem

Abstract

We consider the maximum k-cut problem that consists in partitioning the vertex set of a graph into k subsets such that the sum of the weights of edges joining vertices in different subsets is maximized. We focus on identifying effective classes of inequalities to tighten the semidefinite programming relaxation. We carry out an experimental study of four classes of inequalities from the literature: clique, general clique, wheel and bicycle wheel. We considered 10 combinations of these classes and tested them on both dense and sparse instances for \( k \in \{3,4,5,7\} \). Our computational results suggest that the bicycle wheel and wheel are the strongest inequalities for \( k=3 \), and that for \( k \in \{4,5,7\} \) the wheel inequalities are the strongest by far. Furthermore, we observe an improvement in the performance for all choices of k when both bicycle wheel and wheel are used, at the cost of 72% more CPU time on average when compared with using only one of them.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

References

  1. Anjos, M. F., Ghaddar, B., Hupp, L., Liers, F., & Wiegele, A. (2013). Solving \(k\)-way graph partitioning problems to optimality: The impact of semidefinite relaxations and the bundle method. In M. Jünger & G. Reinelt (Eds.), Facets of combinatorial optimization (pp. 355–386). Berlin: Springer.

    Google Scholar 

  2. Barahona, F., Grötschel, M., Jünger, M., & Reinelt, G. (1988). An application of combinatorial optimization to statistical physics and circuit layout design. Operations Research, 36(3), 493–513.

    Article  Google Scholar 

  3. Chopra, S., & Rao, M. R. (1993). The partition problem. Mathematical Programming, 59(1), 87–115.

    Article  Google Scholar 

  4. Chopra, S., & Rao, M. R. (1995). Facets of the k-partition polytope. Discrete Applied Mathematics, 61(1), 27–48.

    Article  Google Scholar 

  5. Coja-Oghlan, A., Moore, C., & Sanwalani, V. (2006). Max \(k\)-cut and approximating the chromatic number of random graphs. Random Structures and Algorithms, 28(3), 289–322.

    Article  Google Scholar 

  6. Dai, W.-M., & Kuh, E. S. (1987). Simultaneous floor planning and global routing for hierarchical building-block layout. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 6(5), 828–837.

    Article  Google Scholar 

  7. de Klerk, E., Pasechnik, D. V., & Warners, J. P. (2004). On approximate graph colouring and max-\(k\)-cut algorithms based on the \(\theta \)-function. Journal of Combinatorial Optimization, 8(3), 267–294.

    Article  Google Scholar 

  8. Deza, M., Grötschel, M., & Laurent, M. (1992). Clique-web facets for multicut polytopes. Mathematics of Operations Research, 17(4), 981–1000.

    Article  Google Scholar 

  9. Deza, M. M., & Laurent, M. (1997). Geometry of cuts and metrics (1st ed.). Berlin: Springer.

    Google Scholar 

  10. Dolan, E. D., & Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical Programming, 91(2), 201–213.

    Article  Google Scholar 

  11. Eisenblätter, A. (2002). The semidefinite relaxation of the k-partition polytope is strong, volume 2337 of lecture notes in computer science (pp. 273–290). Berlin: Springer.

  12. Fairbrother, J., & Letchford, N. (2016). Projection results for the k-partition problem. Technical Report Optimization Online 5370, Department of Management Science, Lancaster University, UK.

  13. Feo, T. A., & Resende, M. G. C. (1995). Greedy randomized adaptive search procedures. Journal of Global Optimization, 6(2), 109–133.

    Article  Google Scholar 

  14. Frieze, A., & Jerrum, M. (1997). Improved approximation algorithms for maxk-cut and max bisection. Algorithmica, 18(1), 67–81.

    Article  Google Scholar 

  15. Gaur, D., Krishnamurti, R., & Kohli, R. (2008). The capacitated max \(k\)-cut problem. Mathematical Programming, 115(1), 65–72.

    Article  Google Scholar 

  16. Gerards, A. M. H. (1985). Testing the odd bicycle wheel inequalities for the bipartite subgraph polytope. Mathematics of Operations Research, 10(2), 359–360.

    Article  Google Scholar 

  17. Ghaddar, B., Anjos, M. F., & Liers, F. (2011). A branch-and-cut algorithm based on semidefinite programming for the minimum \(k\)-partition problem. Annals of Operations Research, 188(1), 155–174.

    Article  Google Scholar 

  18. Goemans, M. X., & Williamson, D. P. (1995). Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42(6), 1115–1145.

    Article  Google Scholar 

  19. Krislock, N., Malick, J., & Roupin, F. (2012). Improved semidefinite bounding procedure for solving max-cut problems to optimality. Mathematical Programming, 143(1), 61–86.

    Google Scholar 

  20. Liers, F., Jünger, M., Reinelt, G., & Rinaldi, G. (2005). Computing Exact Ground states of hard Ising spin glass problems by branch-and-cut (pp. 47–69). Hoboken: Wiley.

    Google Scholar 

  21. Ma, F., & Hao, J.-K. (2017). A multiple search operator heuristic for the max-k-cut problem. Annals of Operations Research, 248(1), 365–403. doi:10.1007/s10479-016-2234-0.

  22. Mitchell, J. E. (2003). Realignment in the national football league: Did they do it right? Naval Research Logistics, 50(7), 683–701.

    Article  Google Scholar 

  23. Moré, J. J., & Wild, S. M. (2009). Benchmarking derivative-free optimization algorithms. SIAM Journal on Optimization, 20(1), 172–191.

    Article  Google Scholar 

  24. Mosek ApS. (2015).mosek. http://www.mosek.com.

  25. Nikiforov, V. (2016). Max k-cut and the smallest eigenvalue. Linear Algebra and its Applications, 504, 462–467.

    Article  Google Scholar 

  26. Palagi, L., Piccialli, V., Rendl, F., Rinaldi, G., & Wiegele, A. (2011). computational approaches to max-cut. In M. F. Anjos & J. B. Lasserre (Eds.), Handbook of semidefinite, conic and polynomial optimization: Theory, algorithms, software and applications, international series in operations research and management science. New York: Springer.

    Google Scholar 

  27. Papadimitriou, C. H., & Yannakakis, M. (1991). Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43(3), 425–440.

    Article  Google Scholar 

  28. Rendl, F., Rinaldi, G., & Wiegele, A. (2010). Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Mathematical Programming, 121(2), 307–335.

    Article  Google Scholar 

  29. Rinaldi, G. Rudy, a graph generator. https://www-user.tu-chemnitz.de/~helmberg/sdp_software.html.

  30. Scholvin, J. K. (1999). Approximating the longest path problem with heuristics: A survey. Master’s thesis, University of Illinois at Chicago.

  31. Seyed, M. H., Sai, H. T., & Omid, M. (2014). A genetic algorithm for optimization of integrated scheduling of cranes, vehicles, and storage platforms at automated container terminals. Journal of Computational and Applied Mathematics, 270, 545–556. (Fourth international conference on finite element methods in engineering and sciences (FEMTEC 2013)).

    Article  Google Scholar 

  32. Sotirov, R. (2014). An efficient semidefinite programming relaxation for the graph partition problem. INFORMS Journal on Computing, 26(1), 16–30.

    Article  Google Scholar 

  33. van Dam, E. R., & Sotirov, R. (2015). Semidefinite programming and eigenvalue bounds for the graph partition problem. Mathematical Programming, 151(2), 379–404.

    Article  Google Scholar 

  34. van Dam, E. R., & Sotirov, R. (2016). New bounds for the max-k-cut and chromatic number of a graph. Linear Algebra and its Applications, 488, 216–234.

    Article  Google Scholar 

  35. Wiegele, A. (2015). Biq mac library-binary quadratic and max cut library. http://biqmac.uni-klu.ac.at/biqmaclib.html.

Download references

Acknowledgements

This research was supported by Discovery Grants 312125 (M.F. Anjos) and 418250 (S. Le Digabel) from the Natural Sciences and Engineering Research Council of Canada. Moreover, we thank three anonymous reviewers who helped us to significantly improve this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vilmar Jefté Rodrigues de Sousa.

Additional information

We dedicate this paper to Tamás Terlaky on the occasion of his 60th birthday.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rodrigues de Sousa, V.J., Anjos, M.F. & Le Digabel, S. Computational study of valid inequalities for the maximum k-cut problem. Ann Oper Res 265, 5–27 (2018). https://doi.org/10.1007/s10479-017-2448-9

Download citation

Keywords

  • Maximum k-cut
  • Graph partitioning
  • Semidefinite programming
  • Computational study

Mathematics Subject Classification

  • 65K05
  • 90C22
  • 90C35