On February 1, 2023, Benjamin Bergougnoux from the University of Warsaw gave an online talk at the Virtual Discrete Math Colloquium about proving tight lower bounds for the running time of various problems parameterized by rank-width. The title of his talk was “Tight Lower Bounds for Problems Parameterized by Rank-width“.
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.
This is a joint work with Tuukka Korhonen and Jesper Nederlof.
Accepted to STACS 2023 and available on arXiv https://arxiv.org/abs/2210.02117
