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An Experimental Evaluation of Semidefinite Programming and Spectral Algorithms for Max Cut

Authors Renee Mirka, David P. Williamson

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Author Details

Renee Mirka
  • Cornell University, Ithaca, NY, USA
David P. Williamson
  • Cornell University, Ithaca, NY, USA

Funding

This work was supported by the National Science Foundation [NSF CCF-2007009].

Cite AsGet BibTex

Renee Mirka and David P. Williamson. An Experimental Evaluation of Semidefinite Programming and Spectral Algorithms for Max Cut. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SEA.2022.19

Abstract

We experimentally evaluate the performance of several Max Cut approximation algorithms. In particular, we compare the results of the Goemans and Williamson algorithm using semidefinite programming with Trevisan’s algorithm using spectral partitioning. The former algorithm has a known .878 approximation guarantee whereas the latter has a .614 approximation guarantee. We investigate whether this gap in approximation guarantees is evident in practice or whether the spectral algorithm performs as well as the SDP. We also compare the performances to the standard greedy Max Cut algorithm which has a .5 approximation guarantee and two additional spectral algorithms. The algorithms are tested on Erdős-Renyi random graphs, complete graphs from TSPLIB, and real-world graphs from the Network Repository. We find, unsurprisingly, that the spectral algorithms provide a significant speed advantage over the SDP. In our experiments, the spectral algorithms return cuts with values which are competitive with those of the SDP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Max Cut
  • Approximation Algorithms

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References

  1. Francisco Barahona, Martin Grötschel, Michael Jünger, and Gerhard Reinelt. An application of combinatorial optimization to statistical physics and circuit layout design. Operations Research, 36(3):493-513, 1988. URL: https://doi.org/10.1287/opre.36.3.493.
  2. Jonathan W. Berry and Mark K. Goldberg. Path optimization for graph partitioning problems. Discrete Appl. Math., 90(1–3):27-50, January 1999. URL: https://doi.org/10.1016/S0166-218X(98)00084-5.
  3. Alberto Bertoni, Paola Campadelli, and Giuliano Grossi. An approximation algorithm for the maximum cut problem and its experimental analysis. Discrete Applied Mathematics, 110:3-12, 2001. URL: https://doi.org/10.1016/S0166-218X(00)00299-7.
  4. F. Della Croce, M.J. Kaminski, and V.Th. Paschos. An exact algorithm for MAX-CUT in sparse graphs. Operations Research Letters, 35(3):403-408, 2007. URL: https://doi.org/10.1016/j.orl.2006.04.001.
  5. Oliver Dolezal, Thomas Hofmeister, and Hanno Lefmann. A comparison of approximation algorithms for the maxcut-problem, May 2000. URL: https://doi.org/10.17877/DE290R-5013.
  6. Iain Dunning, Swati Gupta, and John Silberholz. What works best when? A systematic evaluation of heuristics for max-cut and QUBO. INFORMS Journal on Computing, 30(3):608-624, 2018. URL: https://doi.org/10.1287/ijoc.2017.0798.
  7. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115-1145, November 1995. URL: https://doi.org/10.1145/227683.227684.
  8. Alexander Golovnev. New upper bounds for MAX-2-SAT and MAX-2-CSP w.r.t. the average variable degree. In Dániel Marx and Peter Rossmanith, editors, Parameterized and Exact Computation, pages 106-117, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. Google Scholar
  9. F. Hadlock. Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing, 4(3):221-225, 1975. URL: https://doi.org/10.1137/0204019.
  10. Timotej Hrga, Borut Lužar, Janez Povh, and Angelika Wiegele. BiqBin: Moving boundaries for NP-hard problems by HPC. In Ivan Dimov and Stefka Fidanova, editors, Advances in High Performance Computing, pages 327-339, Cham, 2021. Springer International Publishing. Google Scholar
  11. Timotej Hrga and Janez Povh. MADAM: A parallel exact solver for max-cut based on semidefinite programming and ADMM. Comput. Optim. Appl., 80(2):347-375, November 2021. URL: https://doi.org/10.1007/s10589-021-00310-6.
  12. Richard Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, volume 40, pages 85-103, January 1972. URL: https://doi.org/10.1007/978-3-540-68279-0_8.
  13. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput., 37(1):319-357, April 2007. URL: https://doi.org/10.1137/S0097539705447372.
  14. Nathan Krislock, Jérôme Malick, and Frédéric Roupin. BiqCrunch: A semidefinite branch-and-bound method for solving binary quadratic problems. ACM Trans. Math. Softw., 43(4), January 2017. URL: https://doi.org/10.1145/3005345.
  15. G.I. Orlova and Ya. G. Dorfman. Finding the maximal cut in a graph. Engineering Cybernetics, 10(3):502-506, 1972. Google Scholar
  16. Jan Poland and Thomas Zeugmann. Clustering pairwise distances with missing data: Maximum cuts versus normalized cuts. In Ljupčo Todorovski, Nada Lavrač, and Klaus P. Jantke, editors, Discovery Science, pages 197-208, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg. Google Scholar
  17. Svatopluk Poljak and Zsolt Tuza. Maximum cuts and largest bipartite subgraphs. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 20, pages 181-244, 1995. URL: https://doi.org/10.1090/dimacs/020/04.
  18. Gerhard Reinelt. TSPLIB - A traveling salesman problem library. ORSA Journal on Computing, 3(4):376-384, 1991. Google Scholar
  19. Franz Rendl, Giovanni Rinaldi, and Angelika Wiegele. Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Mathematical Programming, 121:307-335, February 2010. URL: https://doi.org/10.1007/s10107-008-0235-8.
  20. Ryan A. Rossi and Nesreen K. Ahmed. The network data repository with interactive graph analytics and visualization. In AAAI, 2015. URL: https://networkrepository.com.
  21. Sartaj Sahni and Teofilo Gonzalez. P-Complete approximation problems. J. ACM, 23(3):555-565, July 1976. URL: https://doi.org/10.1145/321958.321975.
  22. José A. Soto. Improved analysis of a max-cut algorithm based on spectral partitioning. SIAM Journal on Discrete Mathematics, 29(1):259-268, 2015. URL: https://doi.org/10.1137/14099098X.
  23. Luca Trevisan. Max cut and the smallest eigenvalue. SIAM Journal on Computing, 41(6):1769-1786, 2012. URL: https://doi.org/10.1137/090773714.
  24. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2):357-365, 2005. Automata, Languages and Programming: Algorithms and Complexity (ICALP-A 2004). URL: https://doi.org/10.1016/j.tcs.2005.09.023.
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