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Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization

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Abstract

A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [Discrete Applied Mathematics, 2020], we introduce a natural generalization of this problem, which we call d -Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d -Cut on \((2d+2)\)-regular graphs and a polynomial algorithm for graphs of maximum degree at most \(d+2\). The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [DAM, 2020], is a polynomial kernel for d -Cut for every positive integer d, parameterized by the vertex deletion distance of the input graph to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time \(\mathcal {O}^*\!\left( 2^n\right)\).

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Notes

  1. The \(\textsf {ETH}\) states that 3-SAT on n variables cannot be solved in time \(2^{o(n)}\); see [17] for more details.

  2. The \(\mathcal {O}^*\!\left( \cdot \right)\) notation suppresses factors that are bounded by a polynomial in the input size.

References

  1. Aravind, N.R., Kalyanasundaram, S., Kare, A.S.: On structural parameterizations of the matching cut problem. In: Proceedings of the 11th International Conference on Combinatorial Optimization and Applications (COCOA), Vol. 10628 of LNCS, pp. 475–482 (2017)

  2. Ban, A., Linial, N.: Internal partitions of regular graphs. J. Graph Theory 83(1), 5–18 (2016)

    Article  MathSciNet  Google Scholar 

  3. Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    Article  MathSciNet  Google Scholar 

  4. Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS), Vol. 9 of LIPIcs, pp. 165–176 (2011)

  5. Bonsma, P.S.: The complexity of the matching-cut problem for planar graphs and other graph classes. J. Graph Theory 62(2), 109–126 (2009)

    Article  MathSciNet  Google Scholar 

  6. Boral, A., Cygan, M., Kociumaka, T., Pilipczuk, M.: A fast branching algorithm for cluster vertex deletion. Theory Comput. Syst. 58(2), 357–376 (2016)

    Article  MathSciNet  Google Scholar 

  7. Chvátal, V.: Recognizing decomposable graphs. J. Graph Theory 8(1), 51–53 (1984)

    Article  MathSciNet  Google Scholar 

  8. Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

    Article  MathSciNet  Google Scholar 

  9. Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)

    Book  Google Scholar 

  10. DeVos, M.: http://www.openproblemgarden.org/op/friendly\_partitions (2009)

  11. Diestel, R.: Graph Theory, vol. 173, 4th edn. Springer, Berlin (2010)

    Book  Google Scholar 

  12. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Berlin (2013)

    Book  Google Scholar 

  13. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer, Berlin (2010)

    Book  Google Scholar 

  14. Fomin, F.V., Lokshtanov, D., Saurabh, S., Zehavi, M.: Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, Cambridge (2019)

    MATH  Google Scholar 

  15. Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)

    Article  MathSciNet  Google Scholar 

  16. Graham, R.L.: On primitive graphs and optimal vertex assignments. Ann. New York Acad. Sci. 175(1), 170–186 (1970)

    Article  MathSciNet  Google Scholar 

  17. Impagliazzo, R., Paturi, R.: On the complexity of \(k\)-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)

    Article  MathSciNet  Google Scholar 

  18. Kaneko, A.: On decomposition of triangle-free graphs under degree constraints. J. Graph Theory 27(1), 7–9 (1998)

    Article  MathSciNet  Google Scholar 

  19. Kloks, T.: Treewidth. Computations and Approximations. Springer, Berlin (1994)

    MATH  Google Scholar 

  20. Komusiewicz, C., Kratsch, D., Le, V.B.: Matching cut: Kernelization, single-exponential time fpt, and exact exponential algorithms. 283, 44–58 (2020)

  21. Kratsch, D., Le, V.B.: Algorithms solving the matching cut problem. Theo. Comput. Sci. 609, 328–335 (2016)

    Article  MathSciNet  Google Scholar 

  22. Le, H., Le, V.B.: On the complexity of matching cut in graphs of fixed diameter. In: Proceedings of the 27th International Symposium on Algorithms and Computation (ISAAC), Vol. 64 of LIPIcs, pp. 50:1–50:12 (2016)

  23. Le, V.B., Randerath, B.: On stable cutsets in line graphs. Theo. Comput. Sci. 301(1–3), 463–475 (2003)

    Article  MathSciNet  Google Scholar 

  24. Lovász, L.: Coverings and colorings of hypergraphs. In: Proceedings of the 4th Southeastern Conference of Combinatorics, Graph Theory, and Computing, pp. 3–12. Utilitas Mathematica Publishing (1973)

  25. Ma, J., Yang, T.: Decomposing \(C_4\)-free graphs under degree constraints. J. Graph Theory 90(1), 13–23 (2019)

    Article  MathSciNet  Google Scholar 

  26. Marx, D., O’sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. ACM Trans. Algorithms, 9(4) (2013)

  27. Moshi, A.M.: Matching cutsets in graphs. J. Graph Theory 13(5), 527–536 (1989)

    Article  MathSciNet  Google Scholar 

  28. M. Patrignani, M., Pizzonia, M.: The complexity of the matching-cut problem. In: Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), Vol. 2204 of LNCS, pp. 284–295 (2001)

  29. Robertson, N., Seymour, P.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)

    Article  MathSciNet  Google Scholar 

  30. K. H. Shafique, K. H., Dutton, R. D.: On satisfactory partitioning of graphs. Congressus Numerantium, pp 183–194, (2002)

  31. Stiebitz, M.: Decomposing graphs under degree constraints. J. Graph Theory 23(3), 321–324 (1996)

    Article  MathSciNet  Google Scholar 

  32. Thomassen, C.: Graph decomposition with constraints on the connectivity and minimum degree. J. Graph Theory 7(2), 165–167 (1983)

    Article  MathSciNet  Google Scholar 

  33. Yap, C.: Some consequences of non-uniform conditions on uniform classes. Theo. Comput. Sci. 26, 287–300 (1983)

    Article  Google Scholar 

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Acknowledgements

We would like to thank the anonymous reviewers for their very pertinent and thorough remarks that improved the presentation of the manuscript.

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Authors and Affiliations

  1. Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

    Guilherme C. M. Gomes

  2. LIRMM, Université de Montpellier, Montpellier, France

    Guilherme C. M. Gomes

  3. LIRMM, Université de Montpellier, CNRS, Montpellier, France

    Ignasi Sau

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Correspondence to Guilherme C. M. Gomes.

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G.C.M. Gomes was funded by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. I. Sau was funded by Projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010). This article is permanently available at [arXiv:1905.03134]. A conference version appeared in the Proc. of the 14th International Symposium on Parameterized and Exact Computation (IPEC), Munich, Germany, September 2019.

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Gomes, G.C.M., Sau, I. Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization. Algorithmica 83, 1677–1706 (2021). https://doi.org/10.1007/s00453-021-00798-8

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