Skip to main content
SpringerLink
Log in

A New Global Algorithm for Max-Cut Problem with Chordal Sparsity

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we develop a semidefinite relaxation-based branch-and-bound algorithm that exploits the chordal sparsity patterns of the max-cut problem. We first study how the chordal sparsity pattern affects the hardness of a max-cut problem. To do this, we derive a polyhedral relaxation based on the clique decomposition of the chordal sparsity patterns and prove some sufficient conditions for the tightness of this polyhedral relaxation. The theoretical results show that the max-cut problem is easy to solve when the sparsity pattern embedded in the problem has a small treewidth and the number of vertices in the intersection of maximal cliques is small. Based on the theoretical results, we propose a new branching rule called hierarchy branching rule, which utilizes the tree decomposition of the sparsity patterns. We also analyze how the proposed branching rule affects the chordal sparsity patterns embedded in the problem, and explain why it can be effective. The numerical experiments show that the proposed algorithm is superior to those known algorithms using classical branching rules and the state-of-the-art solver BiqCrunch on most instances with sparsity patterns arisen in practical applications.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data availability statement

The datasets generated during the computational experiments are available in the Github repository: https://github.com/zhibindeng/Personal/blob/gh-pages/TestSet.zip.

Notes

  1. It is easy to check that Properties (P3) and (P3’) are equivalent.

  2. See https://biqcrunch.lipn.univ-paris13.fr/BiqCrunch/results for detailed numerical results of BiqCrunch and Biq Mac.

  3. Available at https://biqcrunch.lipn.univ-paris13.fr/BiqCrunch. Our results is based on the second release of BiqCrunch.

  4. See http://biqmac.uni-klu.ac.at/biqmaclib.html.

  5. We generated instances with different \(k=5,6,7,8\) for the given w and s and found that the proposed algorithm can solve the largest instance for \(k=8\) within 7 min, while the benchmark algorithm R2 already ran out of time limit. Hence, we did not try to find the largest possible instances that can be solved within 3 h by our algorithm for this type of sparsity pattern.

  6. We found that, even for \(n=120\), some instances can not be solved by any of the algorithm within 3 h. Hence, the step of graph augmentation is skipped in generating disk graphs in this set.

References

  1. Andersen, M., Vandenberghe, L., Dahl, J.: Linear matrix inequalities with chordal sparsity patterns and applications to robust quadratic optimization. In: 2010 IEEE International Symposium on Computer-Aided Control System Design (CACSD), pp. 7–12 (2010)

  2. Arnborg, S., Corneil, D., Proskurowski, A.: Complexity of finding embeddings in a \(k\)-tree. SIAM. J. Alg. Disc. Meth. 8(2), 277–284 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barahona, F., Grötschel, M., Mahjoub, A.R.: Facets of the bipartite subgraph polytope. Math. Oper. Res. 10(2), 340–358 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Benati, S., Ponce, D., Puerto, J., Rodriguez-Chia, A.: A branch-and-price procedure for clustering data that are graph connected. Euro. J. Oper. Res. 297(3), 817–830 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Diestel, R.: Graph Theory, 5th edn. Springer (2017)

  9. Dunning, I., Gupta, S., Silberholz, J.: What works best when? A systematic evaluation of heuristics for max-cut and QUBO. INFORMS J. Comput. 30(3), 608–624 (2018)

    Article  MATH  Google Scholar 

  10. Fairbrother, J., Letchford, A.N., Briggs, K.: A two-level graph partitioning problem arising in mobile wireless communications. Comput. Optim. Appl. 69(3), 653–676 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Festa, P., Pardalos, P., Resende, M., Ribeiro, C.: Randomized heuristics for the max-cut problem. Optim. Methods Softw. 17(6), 1033–1058 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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 

  13. Gaar, E., Rendl, F.: A computational study of exact subgraph based SDP bounds for Max-Cut, stable set and coloring. Math. Program. Ser. B 183(1–2), 283–308 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  15. Garstka, M., Cannon, M., Goulart, P.: Cosmo: a conic operator splitting method for convex conic problems. J. Optim. Theory App. 190(3), 779–810 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

  17. Gosz, M.: Finite Element Method: Applications in Solids, Structures, and Heat Transfer. CRC Press, Boca Raton (2017)

    Book  MATH  Google Scholar 

  18. 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 

  19. Heggernes, P.: Minimal triangulation of graphs: a survey. Discrete Math. 306(3), 297–317 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Jarre, F., Lieder, F., Liu, Y., Lu, C.: Set-completely-positive representations and cuts for the max-cut polytope and the unit modulus lifting. J. Glob. Optim. 76(4), 913–932 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jünger, M., Mallach, S.: Exact facetial odd-cycle separation for maximum cut and binary quadratic optimization. INFORMS J. Comput. 33(4), 1419–1430 (2021)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Krislock, N., Malick, J., Roupin, F.: Improved semidefinite bounding procedure for solving max-cut problems to optimality. Math. Program. Ser. A 143(1), 62–86 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Krislock, N., Malick, J., Roupin, F.: Biqcrunch: a semidefinite branch-and-bound method for solving binary quadratic problems. ACM T. Math. Softw. 43(4), 1–23 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lasserre, J.B.: A MAX-CUT formulation of 0/1 programs. Oper. Res. Lett. 44(2), 158–164 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Madani, R., Sojoudi, S., Fazelnia, G., Lavaei, J.: Finding low-rank solutions of sparse linear matrix inequalities using convex optimization. SIAM J. Optim. 27(2), 725–758 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Martí, R., Duarte, A., Laguna, M.: Advanced scatter search for the max-cut problem. INFORMS J. Comput. 21(1), 26–38 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mosek: Mosek aps. http://www.mosek.com (2020)

  31. 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)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. Ser. A 121(2), 307–335 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rinaldi, G.: Rudy. http://www-user.tu-chemnitz.de/ (1998)

  35. Sliwak, J., Andersen, E.D., Anjos, M.F., Létocart, L., Traversi, E.: A clique merging algorithm to solve semidefinite relaxations of optimal power flow problems. IEEE Trans. Power Syst. 36(2), 1641–1644 (2021)

    Article  Google Scholar 

  36. Tarjan, R., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  37. Teng, J., Jakeman, A., Vaze, J., Croke, B., Dutta, D., Kim, S.: Flood inundation modelling: A review of methods, recent advances and uncertainty analysis. Environ. Modell. Softw. 90, 201–216 (2017)

    Article  Google Scholar 

  38. Vandenberghe, L., Andersen, M.S.: Chordal graphs and semidefinite optimization. Found. Trends Optim. 1(4), 241–443 (2014)

    Article  Google Scholar 

  39. 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 

  40. Wang, J., Magron, V.: Exploiting sparsity in complex polynomial optimization. J. Optim. Theory App. 192(1), 335–359 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, J., Magron, V., Lasserre, J.B.: Chordal-TSSOS: a moment-SOS hierarchy that exploits term sparsity with chordal extension. SIAM J. Optim. 31(1), 114–141 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, J., Magron, V., Lasserre, J.B., Mai, N.H.A.: CS-TSSOS: Correlative and term sparsity for large scale polynomial optimization. arXiv:2005.02828 (2021)

  43. Zhang, R.Y., Lavaei, J.: Sparse semidefinite programs with guaranteed near-linear time complexity via dualized clique tree conversion. Math. Program. Ser. A 188(1), 351–393 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zheng, Y., Fantuzzi, G., Papachristodoulou, A., Goulart, P., Wynn, A.: Chordal decomposition in operator-splitting methods for sparse semidefinite programs. Math. Program. Ser. A 180(1), 489–532 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Lu’s research has been supported by the National Natural Science Foundation of China Grant No. 12171151. Deng’s research has been supported by the National Natural Science Foundation of China Grant No. T2293774, by the Fundamental Research Funds for the Central Universities E2ET0808X2, and by a grant from MOE Social Science Laboratory of Digital Economic Forecast and Policy Simulation at UCAS. Fang’s research has been supported by the Walter Clark Endowment at NC State. Xing’s research has been supported by the National Natural Science Foundation of China Grant No. 11771243.

Author information

Authors and Affiliations

  1. School of Economics and Management, North China Electric Power University, Beijing, 102206, China

    Cheng Lu

  2. School of Economics and Management, University of Chinese Academy of Sciences; MOE Social Science Laboratory of Digital Economic Forecasts and Policy Simulation at UCAS, Beijing, 100190, China

    Zhibin Deng

  3. Department of Industrial and System Engineering, North Carolina State University, Raleigh, 27695-7906, USA

    Shu-Cherng Fang

  4. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Wenxun Xing

Authors
  1. Cheng Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhibin Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shu-Cherng Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wenxun Xing
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhibin Deng.

Additional information

Communicated by Bernard Ries.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, C., Deng, Z., Fang, SC. et al. A New Global Algorithm for Max-Cut Problem with Chordal Sparsity. J Optim Theory Appl 197, 608–638 (2023). https://doi.org/10.1007/s10957-023-02195-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02195-3

Keywords

Mathematics Subject Classification

Search

Navigation