Tight Lower Bounds for Problems Parameterized by Rank-width
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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 Kanté [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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