Tight Lower Bounds for Problems Parameterized by Rank-width

TL;DR

This paper establishes tight lower bounds under ETH for several graph problems parameterized by rank-width, matching existing algorithms and resolving open questions about their optimality.

Contribution

It provides the first tight ETH lower bounds for problems parameterized by rank-width, confirming the optimality of known algorithms and answering open research questions.

Findings

No $2^{o(k^2)} n^{O(1)}$ algorithm exists for Independent Set unless ETH fails.

Known $2^{O(k^2)} n^{O(1)}$ algorithms for Weighted Dominating Set, Induced Matching, and Feedback Vertex Set are optimal.

First tight ETH lower bounds for problems parameterized by rank-width that are not derived from bounds on $n$-vertex graphs.

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.