Parameterized Complexity of Perfectly Matched Sets
For an undirected graph G, a pair of vertex disjoint subsets (A, B) is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (= |B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2^O(√k)⋅ n^O(1), and ii) K_{b,b}-free graphs. We obtain a linear kernel for planar graphs and k^𝒪(d)-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.
Perfectly Matched Sets
Parameterized Complexity
Apex-minor-free graphs
d-degenerate graphs
Planar graphs
Interval Graphs
Theory of computation~Fixed parameter tractability
2:1-2:13
Regular Paper
Akanksha
Agrawal
Akanksha Agrawal
Indian Institute of Technology Madras, Chennai, India
https://orcid.org/0000-0002-0656-7572
Supported by New Faculty Initiation Grant no. NFIG008972.
Sutanay
Bhattacharjee
Sutanay Bhattacharjee
Indian Institute of Technology Madras, Chennai, India
Satyabrata
Jana
Satyabrata Jana
The Institute of Mathematical Sciences, HBNI, Chennai, India
https://orcid.org/0000-0002-7046-0091
Abhishek
Sahu
Abhishek Sahu
Indian Institute of Technology Madras, Chennai, India
10.4230/LIPIcs.IPEC.2022.2
N. R. Aravind and Roopam Saxena. Perfectly Matched Sets in Graphs: Hardness, Kernelization Lower Bound, and FPT and Exact Algorithms. CoRR, abs/2107.08584, 2021. URL: http://arxiv.org/abs/2107.08584.
http://arxiv.org/abs/2107.08584
Paul S. Bonsma. The complexity of the matching-cut problem for planar graphs and other graph classes. J. Graph Theory, 62(2):109-126, 2009.
Leizhen Cai, Siu Man Chan, and Siu On Chan. Random Separation: A New Method for Solving Fixed-Cardinality Optimization Problems. In Hans L. Bodlaender and Michael A. Langston, editors, Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, volume 4169 of Lecture Notes in Computer Science, pages 239-250. Springer, 2006.
Kathie Cameron. Induced matchings in intersection graphs. Discret. Math., 278(1-3):1-9, 2004.
Yixin Cao and Dániel Marx. Interval Deletion Is Fixed-Parameter Tractable. ACM Trans. Algorithms, 11(3):21:1-21:35, 2015.
Chi-Yeh Chen, Sun-Yuan Hsieh, Hoàng-Oanh Le, Van Bang Le, and Sheng-Lung Peng. Matching Cut in Graphs with Large Minimum Degree. Algorithmica, 83(5):1238-1255, 2021.
Vasek Chvátal. Recognizing decomposable graphs. J. Graph Theory, 8(1):51-53, 1984.
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
Konrad K. Dabrowski, Marc Demange, and Vadim V. Lozin. New results on maximum induced matchings in bipartite graphs and beyond. Theor. Comput. Sci., 478:33-40, 2013.
Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos. The Bidimensional Theory of Bounded-Genus Graphs. SIAM J. Discret. Math., 20(2):357-371, 2006.
Reinhard Diestel. Graph theory. Graduate texts in mathematics, 173, 2017.
Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999.
Shimon Even, Oded Goldreich, Shlomo Moran, and Po Tong. On the NP-completeness of certain network testing problems. Networks, 14(1):1-24, 1984.
Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006.
Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Subexponential algorithms for partial cover problems. Inf. Process. Lett., 111(16):814-818, 2011.
Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. CoRR, abs/1311.3899, 2013. URL: http://arxiv.org/abs/1311.3899.
http://arxiv.org/abs/1311.3899
Stasys Jukna. Extremal Combinatorics - With Applications in Computer Science. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2011.
Iyad A. Kanj, Michael J. Pelsmajer, Marcus Schaefer, and Ge Xia. On the induced matching problem. J. Comput. Syst. Sci., 77(6):1058-1070, 2011.
Johannes Köbler, Sebastian Kuhnert, and Osamu Watanabe. Interval graph representation with given interval and intersection lengths. Journal of Discrete Algorithms, 34:108-117, 2015.
Łukasz Kowalik, Matjaž Krnc, Tomasz Waleń, et al. Improved induced matchings in sparse graphs. Discrete Applied Mathematics, 158(18):1994-2003, 2010.
Hoàng-Oanh Le and Van Bang Le. On the Complexity of Matching Cut in Graphs of Fixed Diameter. In 27th International Symposium on Algorithms and Computation, ISAAC 2016, December 12-14, 2016, Sydney, Australia, volume 64 of LIPIcs, pages 50:1-50:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016.
Van Bang Le and Jan Arne Telle. The Perfect Matching Cut Problem Revisited. In Graph-Theoretic Concepts in Computer Science - 47th International Workshop, WG 2021, Warsaw, Poland, June 23-25, 2021, Revised Selected Papers, volume 12911 of Lecture Notes in Computer Science, pages 182-194. Springer, 2021.
Silvio Micali and Vijay V. Vazirani. An O(sqrt(|v|) |E|) Algorithm for Finding Maximum Matching in General Graphs. In 21st Annual Symposium on Foundations of Computer Science, Syracuse, New York, USA, 13-15 October 1980, pages 17-27. IEEE Computer Society, 1980.
Hannes Moser and Somnath Sikdar. The parameterized complexity of the induced matching problem. Discret. Appl. Math., 157(4):715-727, 2009.
Augustine M. Moshi. Matching cutsets in graphs. J. Graph Theory, 13(5):527-536, 1989.
Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006.
Larry J. Stockmeyer and Vijay V. Vazirani. NP-Completeness of Some Generalizations of the Maximum Matching Problem. Inf. Process. Lett., 15(1):14-19, 1982.
Michele Zito. Induced matchings in regular graphs and trees. In Graph-Theoretic Concepts in Computer Science, pages 89-101, Berlin, Heidelberg, 1999. Springer Berlin Heidelberg.
Akanksha Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, and Abhishek Sahu
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