Parameterized Algorithms for Maximum Cut with Connectivity Constraints

Authors Hiroshi Eto, Tesshu Hanaka , Yasuaki Kobayashi, Yusuke Kobayashi



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Author Details

Hiroshi Eto
  • Kyushu University, Fukuoka, Japan
Tesshu Hanaka
  • Chuo University, Tokyo, Japan
Yasuaki Kobayashi
  • Kyoto University, Kyoto, Japan
Yusuke Kobayashi
  • Kyoto University, Kyoto, Japan

Acknowledgements

We thank Akitoshi Kawamura and Yukiko Yamauchi for giving an opportunity to discuss in the Open Problem Seminar at Kyushu University. This work is partially supported by JST CREST JPMJCR1401 and JSPS KAKENHI Grant Numbers JP17H01788, JP18H06469, JP16K16010, JP17K19960, and JP18H05291.

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Hiroshi Eto, Tesshu Hanaka, Yasuaki Kobayashi, and Yusuke Kobayashi. Parameterized Algorithms for Maximum Cut with Connectivity Constraints. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.IPEC.2019.13

Abstract

We study two variants of Maximum Cut, which we call Connected Maximum Cut and Maximum Minimal Cut, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas Maximum Cut on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-width, and twin-cover number. Finally, we obtain FPT algorithms with respect to the solution size.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Maximum cut
  • Parameterized algorithm
  • NP-hardness
  • Graph parameter

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References

  1. C. Bazgan, L. Brankovic, K. Casel, H. Fernau, K. Jansen, K.-M. Klein, M. Lampis, M. Liedloff, J. Monnot, and V. T. Paschos. The many facets of upper domination. Theoretical Computer Science, 717:2-25, 2018. Google Scholar
  2. E. Birmelé, J. A. Bondy, and B. A. Reed. Brambles, Prisms and Grids, pages 37-44. Birkhäuser Basel, Basel, 2007. Google Scholar
  3. H. L. Bodlaender, M. Cygan, S. Kratsch, and J. Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Information and Computation, 243:86-111, 2015. Google Scholar
  4. H. L. Bodlaender, R. G. Downey, M. R. Fellows, and D. Hermelin. On problems without polynomial kernels. Journal of Computer and System Sciences, 75(8):423-434, 2009. Google Scholar
  5. H. L. Bodlaender, P. G. Drange, M. S. Dregi, F. V. Fomin, D. Lokshtanov, and M. Pilipczuk. A c^k n 5-Approximation Algorithm for Treewidth. SIAM Journal on Computing, 45(2):317-378, 2016. Google Scholar
  6. H. L. Bodlaender, J. R. Gilbert, H. Hafsteinsson, and T. Kloks. Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree. Journal of Algorithms, 18(2):238-255, 1995. Google Scholar
  7. H. L. Bodlaender and K. Jansen. On the Complexity of the Maximum Cut Problem. Nordic Journal of Computing, 7(1):14-31, 2000. Google Scholar
  8. N. Boria, F. D. Croce, and V. T. Paschos. On the max min vertex cover problem. Discrete Applied Mathematics, 196:62-71, 2015. Google Scholar
  9. A. Boyacı, T. Ekim, and M. Shalom. A polynomial-time algorithm for the maximum cardinality cut problem in proper interval graphs. Information Processing Letters, 121:29-33, 2017. Google Scholar
  10. B.-M. Bui-Xuan, O. Suchý, J. A. Telle, and M. Vatshelle. Feedback vertex set on graphs of low clique-width. European Journal of Combinatorics, 34(3):666-679, 2013. Google Scholar
  11. R. Carvajal, M. Constantino, M. Goycoolea, J. P. Vielma, and A. Weintraub. Imposing Connectivity Constraints in Forest Planning Models. Operations Research, 61(4):824-836, 2013. Google Scholar
  12. B. Chaourar. A Linear Time Algorithm for a Variant of the MAX CUT Problem in Series Parallel Graphs. Advances in Operations Research, pages 1267108:1-1267108:4, 2017. Google Scholar
  13. B. Chaourar. Connected max cut is polynomial for graphs without K₅⧵ e as a minor. CoRR, abs/1903.12641, 2019. Google Scholar
  14. B. Courcelle and S. Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1):77-114, 2000. Google Scholar
  15. M. Cygan. Deterministic Parameterized Connected Vertex Cover. In SWAT 2012, pages 95-106, 2012. Google Scholar
  16. M. Cygan, F. V. Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer International Publishing, 2015. Google Scholar
  17. M. Cygan, J. Nederlof, M. Pilipczuk, M. Pilipczuk, J. M. M. van Rooij, and J. O. Wojtaszczyk. Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time. In FOCS 2011, pages 150-159, 2011. Google Scholar
  18. M. Demange. A Note on the Approximation of a Minimum-Weight Maximal Independent Set. Computational Optimization and Applications, 14(1):157-169, 1999. Google Scholar
  19. R. Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  20. J. Díaz and M. Kamiński. MAX-CUT and MAX-BISECTION are NP-hard on unit disk graphs. Theoretical Computer Science, 377(1):271-276, 2007. Google Scholar
  21. M. R. Fellows, D. Lokshtanov, N. Misra, M. Mnich, F. Rosamond, and S. Saurabh. The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number. Theory of Computing Systems, 45(4):822-848, 2009. Google Scholar
  22. F. V. Fomin, P. Golovach, D. Lokshtanov, and S. Saurabh. Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width. SIAM Journal on Computing, 43(5):1541-1563, 2014. Google Scholar
  23. R. Ganian. Improving Vertex Cover as a Graph Parameter. Discrete Mathematics and Theoretical Computer Science, 17(2):77-100, 2015. Google Scholar
  24. M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA, 1979. Google Scholar
  25. M. X. Goemans and D. P. Williamson. Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. Journal of the ACM, 42(6):1115-1145, 1995. Google Scholar
  26. V. Grimm, T. Kleinert, F. Liers, M. Schmidt, and G. Zöttl. Optimal price zones of electricity markets: a mixed-integer multilevel model and global solution approaches. Optimization Methods and Software, 34(2):406-436, 2019. Google Scholar
  27. S. Guha and S. Khuller. Approximation Algorithms for Connected Dominating Sets. Algorithmica, 20(4):374-387, 1998. Google Scholar
  28. V. Guruswami. Maximum cut on line and total graphs. Discrete Applied Mathematics, 92(2):217-221, 1999. Google Scholar
  29. F. Hadlock. Finding a Maximum Cut of a Planar Graph in Polynomial Time. SIAM Journal on Computing, 4(3):221-225, 1975. Google Scholar
  30. D. J. Haglin and S. M. Venkatesan. Approximation and intractability results for the maximum cut problem and its variants. IEEE Transactions on Computers, 40(1):110-113, 1991. Google Scholar
  31. M. T. Hajiaghayi, G. Kortsarz, R. MacDavid, M. Purohit, and K. Sarpatwar. Approximation Algorithms for Connected Maximum Cut and Related Problems. In ESA 2015, pages 693-704, 2015. Google Scholar
  32. T. Hanaka, H. L. Bodlaender, T. C. van der Zanden, and H. Ono. On the Maximum Weight Minimal Separator. In TAMC 2017, pages 304-318, 2017. Google Scholar
  33. P. Hliněný and S. Oum. Finding Branch-Decompositions and Rank-Decompositions. SIAM Journal on Computing, 38(3):1012-1032, 2008. Google Scholar
  34. R. M. Karp. Reducibility among Combinatorial Problems, pages 85-103. Springer US, Boston, MA, 1972. Google Scholar
  35. K. Khoshkhah, M. K. Ghadikolaei, J. Monnot, and F. Sikora. Weighted Upper Edge Cover: Complexity and Approximability. In WALCOM 2019, pages 235-247, 2019. Google Scholar
  36. M. Mahajan and V. Raman. Parameterizing above Guaranteed Values: MaxSat and MaxCut. Journal of Algorithms, 31(2):335-354, 1999. Google Scholar
  37. G. I. Orlova and Y. G. Dorfman. Finding the maximal cut in a graph. Engineering Cyvernetics, 10(3):502-506, 1972. Google Scholar
  38. S. Oum. Approximating Rank-width and Clique-width Quickly. ACM Transactions on Algorithms, 5(1):10:1-10:20, 2008. Google Scholar
  39. S. Oum and P. Seymour. Approximating clique-width and branch-width. Journal of Combinatorial Theory, Series B, 96(4):514-528, 2006. Google Scholar
  40. V. Raman and S. Saurabh. Improved fixed parameter tractable algorithms for two “edge” problems: MAXCUT and MAXDAG. Information Processing Letters, 104(2):65-72, 2007. Google Scholar
  41. M. Rao. Clique-width of graphs defined by one-vertex extensions. Discrete Mathematics, 308(24):6157-6165, 2008. Google Scholar
  42. N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92-114, 1986. Google Scholar
  43. S. Saurabh and M. Zehavi. Parameterized Complexity of Multi-Node Hubs. In IPEC 2018, volume 115, pages 8:1-8:14, 2019. Google Scholar
  44. S. Vicente, V. Kolmogorov, and C. Rother. Graph cut based image segmentation with connectivity priors. In CVPR 2008, pages 1-8, 2008. Google Scholar
  45. M. Yannakakis and F. Gavril. Edge Dominating Sets in Graphs. SIAM Journal on Applied Mathematics, 38(3):364-372, 1980. Google Scholar
  46. M. Zehavi. Maximum Minimal Vertex Cover Parameterized by Vertex Cover. SIAM Journal on Discrete Mathematics, 31(4):2440-2456, 2017. Google Scholar
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