论文标题
紧密的下限，以通过等级宽度参数参数
Tight Lower Bounds for Problems Parameterized by Rank-width
论文作者
论文摘要
我们表明，没有$ 2^{o（k^2）} n^{o（o（1）} $ time算法用于独立设置的$ n $ - vertex图，除非指数时间假设（ETH）失败，否则具有等级宽度$ k $的$ k $ k $。我们的下限匹配$ 2^{o（k^2）} n^{o（1）} $ time算法，由bui-xuan，telle和vatshelle [divet。应用。 Math。，2010年]，它回答了Bergougnoux和Kanté的公开问题[Siam J. Divet。数学，2021年]。我们还表明，已知的$ 2^{o（k^2）} n^{o（1）} $ time算法是加权主导集，最大诱导的匹配和反馈匹配和反馈顶点集，由rank-width $ k $参数为参数是最佳的。我们的结果是第一个按等级宽度参数化的紧密ETH下限，并未直接从$ n $ vertex图的下限直接遵循。
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.
