# A Meta-Algorithm for Solving Connectivity and Acyclicity Constraints on Locally Checkable Properties Parameterized by Graph Width Measures

Abstract : In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that solves all these problems and whose running time is upper bounded by $2^{O(k)}\cdot n^{O(1)}$, $2^{O(k \log(k))}\cdot n^{O(1)}$, $2^{O(k^2)}\cdot n^{O(1)}$ and $n^{O(k)}$ where $k$ is respectively the clique-width, $\mathbb{Q}$-rank-width, rank-width and maximum induced matching width of a given decomposition. Our meta-algorithm simplifies and unifies the known algorithms for each of the parameters and its running time matches asymptotically also the running times of the best known algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the $d$-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance of this equivalence relation on the algorithmic applications of width measures. We also prove that our framework could be useful for $W[1]$-hard problems parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For these latter problems, we obtain $n^{O(k)}$, $n^{O(k)}$ and $n^{2^{O(k)}}$ time algorithms where $k$ is respectively the clique-width, the $\mathbb{Q}$-rank-width and the rank-width of the input graph.
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Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-01799573
Contributor : Benjamin Bergougnoux <>
Submitted on : Thursday, December 5, 2019 - 4:09:33 PM
Last modification on : Monday, January 20, 2020 - 12:14:07 PM

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### Identifiers

• HAL Id : hal-01799573, version 5
• ARXIV : 1805.11275

### Citation

Benjamin Bergougnoux, Mamadou Moustapha Kanté. A Meta-Algorithm for Solving Connectivity and Acyclicity Constraints on Locally Checkable Properties Parameterized by Graph Width Measures. 2019. ⟨hal-01799573v5⟩

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