Separator Based Data Reduction for the Maximum Cut Problem

Authors Jonas Charfreitag , Christine Dahn , Michael Kaibel , Philip Mayer , Petra Mutzel , Lukas Schürmann



PDF
Thumbnail PDF

File

LIPIcs.SEA.2024.4.pdf
  • Filesize: 0.88 MB
  • 21 pages

Document Identifiers

Author Details

Jonas Charfreitag
  • Institute of Computer Science, University of Bonn, Germany
Christine Dahn
  • Institute of Computer Science, University of Bonn, Germany
Michael Kaibel
  • Institute of Computer Science, University of Bonn, Germany
Philip Mayer
  • Institute of Computer Science, University of Bonn, Germany
Petra Mutzel
  • Institute of Computer Science, University of Bonn, Germany
Lukas Schürmann
  • Institute of Computer Science, University of Bonn, Germany

Acknowledgements

We thank Konstantinos Papadopoulos for his contribution to this work.

Cite AsGet BibTex

Jonas Charfreitag, Christine Dahn, Michael Kaibel, Philip Mayer, Petra Mutzel, and Lukas Schürmann. Separator Based Data Reduction for the Maximum Cut Problem. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SEA.2024.4

Abstract

Preprocessing is an important ingredient for solving the maximum cut problem to optimality on real-world graphs. In our work, we derive a new framework for data reduction rules based on vertex separators. Vertex separators are sets of vertices, whose removal increases the number of connected components of a graph. Certain small separators can be found in linear time, allowing for an efficient combination of our framework with existing data reduction rules. Additionally, we complement known data reduction rules for triangles with a new one. In our computational experiments on established benchmark instances, we clearly show the effectiveness and efficiency of our proposed data reduction techniques. The resulting graphs are significantly smaller than in earlier studies and sometimes no vertex is left, so preprocessing has fully solved the instance to optimality. The introduced techniques are also shown to offer significant speedup potential for an exact state-of-the-art solver and to help a state-of-the-art heuristic to produce solutions of higher quality.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Mathematics of computing → Solvers
  • Theory of computation → Network optimization
Keywords
  • Data Reduction
  • Maximum Cut
  • Vertex Separators

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Faisal N. Abu-Khzam, Sebastian Lamm, Matthias Mnich, Alexander Noe, Christian Schulz, and Darren Strash. Recent advances in practical data reduction. In Hannah Bast, Claudius Korzen, Ulrich Meyer, and Manuel Penschuck, editors, Algorithms for Big Data: DFG Priority Program 1736, pages 97-133. Springer Nature Switzerland, Cham, 2022. URL: https://doi.org/10.1007/978-3-031-21534-6_6.
  2. Eugenio Angriman, Alexander van der Grinten, Moritz von Looz, Henning Meyerhenke, Martin Nöllenburg, Maria Predari, and Charilaos Tzovas. Guidelines for experimental algorithmics: A case study in network analysis. Algorithms, 12(7), 2019. URL: https://doi.org/10.3390/a12070127.
  3. Claudio Arbib. A polynomial characterization of some graph partitioning problems. Inf. Process. Lett., 26(5):223-230, 1988. URL: https://doi.org/10.1016/0020-0190(88)90144-5.
  4. Francisco Barahona. The max-cut problem on graphs not contractible to K5. Oper. Res. Lett., 2(3):107-111, August 1983. URL: https://doi.org/10.1016/0167-6377(83)90016-0.
  5. Francisco Barahona, Martin Grötschel, Michael Jünger, and Gerhard Reinelt. An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res., 36(3):493-513, 1988. URL: https://doi.org/10.1287/opre.36.3.493.
  6. Francisco Barahona, Michael Jünger, and Gerhard Reinelt. Experiments in quadratic 0-1 programming. Math. Program., 44(1-3):127-137, 1989. URL: https://doi.org/10.1007/BF01587084.
  7. Samuel Burer, Renato D. C. Monteiro, and Yin Zhang. Rank-two relaxation heuristics for MAX-CUT and other binary quadratic programs. SIAM J. Optim., 12(2):503-521, 2002. URL: https://doi.org/10.1137/S1052623400382467.
  8. Jonas Charfreitag, Michael Jünger, Sven Mallach, and Petra Mutzel. Mcsparse: Exact solutions of sparse maximum cut and sparse unconstrained binary quadratic optimization problems. In Cynthia A. Phillips and Bettina Speckmann, editors, Proceedings of the Symposium on Algorithm Engineering and Experiments, ALENEX 2022, Alexandria, VA, USA, January 9-10, 2022, pages 54-66. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977042.5.
  9. Markus Chimani, Carsten Gutwenger, Michael Jünger, Gunnar W Klau, Karsten Klein, and Petra Mutzel. The open graph drawing framework (ogdf). Handbook of graph drawing and visualization, 2011:543-569, 2013. Google Scholar
  10. Markus Chimani, Martina Juhnke-Kubitzke, Alexander Nover, and Tim Römer. Cut polytopes of minor-free graphs. CoRR, abs/1903.01817, 2019. URL: https://doi.org/10.48550/arXiv.1903.01817.
  11. George B. Dantzig. Discrete-variable extremum problems. Operations Research, 5(2):266-277, 1957. URL: https://doi.org/10.1287/opre.5.2.266.
  12. 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.
  13. Shimon Even and Robert Endre Tarjan. Network Flow and Testing Graph Connectivity. SIAM J. Comput., 4(4):507-518, 1975. URL: https://doi.org/10.1137/0204043.
  14. Damir Ferizovic, Demian Hespe, Sebastian Lamm, Matthias Mnich, Christian Schulz, and Darren Strash. Engineering kernelization for maximum cut. In Guy E. Blelloch and Irene Finocchi, editors, Proceedings of the Symposium on Algorithm Engineering and Experiments, ALENEX 2020, Salt Lake City, UT, USA, January 5-6, 2020, pages 27-41. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611976007.3.
  15. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
  16. 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, 1995. URL: https://doi.org/10.1145/227683.227684.
  17. R. E. Gomory and T. C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9(4):551-570, 1961. URL: https://doi.org/10.1137/0109047.
  18. Martin Grohe. Quasi-4-Connected Components. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 8:1-8:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.8.
  19. Carsten Gutwenger and Petra Mutzel. A linear time implementation of spqr-trees. In Joe Marks, editor, Graph Drawing, 8th International Symposium, GD 2000, Colonial Williamsburg, VA, USA, September 20-23, 2000, Proceedings, volume 1984 of Lecture Notes in Computer Science, pages 77-90. Springer, 2000. URL: https://doi.org/10.1007/3-540-44541-2_8.
  20. F. Hadlock. Finding a maximum cut of a planar graph in polynomial time. SIAM J. Comput., 4(3):221-225, 1975. URL: https://doi.org/10.1137/0204019.
  21. Dorit S. Hochbaum. Why should biconnected components be identified first. Discret. Appl. Math., 42(2):203-210, 1993. URL: https://doi.org/10.1016/0166-218X(93)90046-Q.
  22. J.E. Hopcroft and R.E. Tarjan. Dividing a graph into triconnected components. SIAM Journal on Computing, 2(3):135-158, 1973. URL: https://doi.org/10.1137/0202012.
  23. Falk Hüffner, Nadja Betzler, and Rolf Niedermeier. Separator-based data reduction for signed graph balancing. J. Comb. Optim., 20(4):335-360, 2010. URL: https://doi.org/10.1007/s10878-009-9212-2.
  24. Yoichi Iwata and Takuto Shigemura. Separator-based pruned dynamic programming for steiner tree. In The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27 - February 1, 2019, pages 1520-1527. AAAI Press, 2019. URL: https://doi.org/10.1609/AAAI.V33I01.33011520.
  25. Michael Jünger, Elisabeth Lobe, Petra Mutzel, Gerhard Reinelt, Franz Rendl, Giovanni Rinaldi, and Tobias Stollenwerk. Quantum annealing versus digital computing: An experimental comparison. ACM J. Exp. Algorithmics, 26:1.9:1-1.9:30, 2021. URL: https://doi.org/10.1145/3459606.
  26. Arkady Kanevsky. Finding all minimum-size separating vertex sets in a graph. Networks, 23(6):533-541, 1993. URL: https://doi.org/10.1002/net.3230230604.
  27. Arkady Kanevsky and Vijaya Ramachandran. Improved algorithms for graph four-connectivity. J. Comput. Syst. Sci., 42(3):288-306, 1991. URL: https://doi.org/10.1016/0022-0000(91)90004-O.
  28. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  29. Jan-Hendrik Lange, Bjoern Andres, and Paul Swoboda. Combinatorial persistency criteria for multicut and max-cut. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2019, Long Beach, CA, USA, June 16-20, 2019, pages 6093-6102. Computer Vision Foundation / IEEE, 2019. URL: https://doi.org/10.1109/CVPR.2019.00625.
  30. Frauke Liers and G. Pardella. Partitioning planar graphs: a fast combinatorial approach for max-cut. Comput. Optim. Appl., 51(1):323-344, 2012. URL: https://doi.org/10.1007/s10589-010-9335-5.
  31. Andrea Lodi and Andrea Tramontani. Performance variability in mixed-integer programming. In Theory driven by influential applications, pages 1-12. INFORMS, 2013. URL: https://doi.org/10.1287/educ.2013.0112.
  32. S. Thomas McCormick, M. R. Rao, and Giovanni Rinaldi. Easy and difficult objective functions for max cut. Math. Program., 94(2-3):459-466, 2003. URL: https://doi.org/10.1007/s10107-002-0328-8.
  33. Hans Mittelmann. Qubo benchmark. Website, 2023. Accessed on 2023-08-07. URL: https://plato.asu.edu/ftp/qubo.html.
  34. Christos H. Papadimitriou and Mihalis Yannakakis. Optimization, approximation, and complexity classes. J. Comput. Syst. Sci., 43(3):425-440, 1991. URL: https://doi.org/10.1016/0022-0000(91)90023-X.
  35. Daniel Rehfeldt, Thorsten Koch, and Yuji Shinano. Faster exact solution of sparse maxcut and qubo problems. Mathematical Programming Computation, pages 1-26, 2023. URL: https://doi.org/10.1007/s12532-023-00236-6.
  36. Ryan A. Rossi and Nesreen K. Ahmed. The network data repository with interactive graph analytics and visualization. In Blai Bonet and Sven Koenig, editors, Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, January 25-30, 2015, Austin, Texas, USA, pages 4292-4293. AAAI Press, 2015. URL: https://doi.org/10.1609/aaai.v29i1.9277.
  37. Carsten Rother, Vladimir Kolmogorov, Victor S. Lempitsky, and Martin Szummer. Optimizing binary mrfs via extended roof duality. In 2007 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2007), 18-23 June 2007, Minneapolis, Minnesota, USA. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/CVPR.2007.383203.
  38. Christian L. Staudt, Aleksejs Sazonovs, and Henning Meyerhenke. Networkit: A tool suite for large-scale complex network analysis. Netw. Sci., 4(4):508-530, 2016. URL: https://doi.org/10.1017/NWS.2016.20.
  39. R. Tamassia and G. Di Battista. Incremental planarity testing. In 30th Annual Symposium on Foundations of Computer Science, pages 436-441, Los Alamitos, CA, USA, November 1989. IEEE Computer Society. URL: https://doi.org/10.1109/SFCS.1989.63515.
  40. Robert Endre Tarjan. Depth-first search and linear graph algorithms. SIAM J. Comput., 1(2):146-160, 1972. URL: https://doi.org/10.1137/0201010.
  41. K Wagner. Beweis einer Abschwächung der Hadwiger-Vermutung. Mathematische Annalen, 153(2):139-141, 1964. Google Scholar
  42. Angelika Wiegele. Biq Mac Library - a collection of Max-Cut and quadratic 0-1 programming instances of medium size. http://biqmac.aau.at/biqmaclib.html, 2009.
  43. Thomas Victor Wimer. Linear Algorithms on K-Terminal Graphs. PhD thesis, Clemson University, USA, 1987. AAI8803914. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail