# P-FPT algorithms for bounded clique-width graphs

1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as the Strong Exponential Time Hypothesis. According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms. A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms with polynomial dependency in the fixed parameter (P-FPT). Applying this technique to clique-width, an important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems, distance problems and maximum matching. We give hardness results and P-FPT algorithms, using clique-width and some of its upper bounds as parameters. We believe that our most important result is an algorithm in $O(k^4 · n + m)$-time for computing a maximum matching where $k$ is either the modular-width of the graph or the $P_4$-sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs. Our algorithms are based on preprocessing methods using modular decomposition and split decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width.
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https://hal.archives-ouvertes.fr/hal-02152971
Contributor : Guillaume Ducoffe <>
Submitted on : Tuesday, June 11, 2019 - 6:28:51 PM
Last modification on : Tuesday, May 26, 2020 - 6:50:53 PM

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### Citation

David Coudert, Guillaume Ducoffe, Alexandru Popa. P-FPT algorithms for bounded clique-width graphs. ACM Transactions on Algorithms, Association for Computing Machinery, 2019, 15 (3), pp.1-57. ⟨10.1145/3310228⟩. ⟨hal-02152971⟩

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