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arXiv · 2210.02117

Tight Lower Bounds for Problems Parameterized by Rank-width

Benjamin Bergougnoux·Tuukka Korhonen·Jesper Nederlof
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Abstract

We show that there is no $2^{o(k^2)} n^{O(1)}$ time algorithm for Independent Set on $n$-vertex graphs with rank-width $k$, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the $2^{O(k^2)} n^{O(1)}$ time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kant\'{e} [SIAM J. Discret. Math., 2021]. We also show that the known $2^{O(k^2)} n^{O(1)}$ time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width $k$ are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for $n$-vertex graphs.

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Benjamin Bergougnoux, Tuukka Korhonen, Jesper Nederlof. 2022-10-05. Tight Lower Bounds for Problems Parameterized by Rank-width. https://arxiv.org/abs/2210.02117

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