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.

翻译：我们显示,单元值为1美元(k)2美元(n ⁇ 1美元)的顶点图上,没有2美元(k ⁇ 2)的美元时间算法,除非光学时间假设(ETH)失败。我们的下限匹配了布伊-Xuan、Telle和Vatshelle[Discret.Appl. Math., 2010]给出的2美元(k ⁇ 2)的时间算法。它回答了Bergougnoux和Kant\{e}[SIAM J. Discret. Math., 2021][SIAM. Discret. Math., 20211] 的开放问题。我们还显示,已知的2美元(k)2美元(k ⁇ 2)}(n ⁇ 1)的时间算法,即按下维特美元参数设定的加权值最大匹配和反馈Vertex设置的时间算得最优。我们的结果是假设ET。第一个按标准维度测量的较窄的埃塞俄比亚较低界限参数,不是直接从$-verfex图表的下框。