对参数化复杂性的精确结构阈值

Falko Hegerfeld, Stefan Kratsch
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引用次数: 3

摘要

参数化复杂性寻求使用输入结构来获得np困难问题的更快算法。这对于低树宽的图来说是最成功的:许多问题允许相对于树宽的快速算法,其中许多算法在SETH下是最优的。对于更一般的结构(如低团宽度)和更严格的结构(如到稀疏图类的低删除距离),这样的结果较少。尽管取得了这些成功,但这些结果在可能的结构范围内仍然是“孤岛”。而不是增加更多的孤岛,我们试图确定它们之间的转换,也就是说,我们的目标是结构阈值,其中复杂性随着输入结构变得更加普遍而增加。从删除距离到树宽,单个删除设置到具有简单组件的图是否足以产生与树宽相同的下界,或者是否需要许多不相交的分隔符?从树的宽度到团的宽度,多少密度需要与团的宽度相同的复杂性?相反,产生相同下界的最严格的结构是什么?对于treewidth,我们获得了已经适用于具有单个分隔符$X$的图的改进和新的下界,使得$G-X$具有树宽$r=O(1)$,而$G$具有树宽$|X|+O(1)$。我们排除了在$O^*((r+1-\epsilon)^{k})$时间内运行的算法,用于删除$r$-可着色的参数为$k=|X|$。对于clique-width,我们排除了$O^*((2^r-\epsilon)^k)$用于删除$r$- colorable,其中$X$现在允许由$k$双类组成。在顶点覆盖、支配集和最大切割上有进一步的结果。所有下界都由现有的和新设计的算法匹配。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Towards exact structural thresholds for parameterized complexity
Parameterized complexity seeks to use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth: Many problems admit fast algorithms relative to treewidth and many of them are optimal under SETH. Fewer such results are known for more general structure such as low clique-width and more restrictive structure such as low deletion distance to a sparse graph class. Despite these successes, such results remain"islands'' within the realm of possible structure. Rather than adding more islands, we seek to determine the transitions between them, that is, we aim for structural thresholds where the complexity increases as input structure becomes more general. Going from deletion distance to treewidth, is a single deletion set to a graph with simple components enough to yield the same lower bound as for treewidth or does it take many disjoint separators? Going from treewidth to clique-width, how much more density entails the same complexity as clique-width? Conversely, what is the most restrictive structure that yields the same lower bound? For treewidth, we obtain both refined and new lower bounds that apply already to graphs with a single separator $X$ such that $G-X$ has treewidth $r=O(1)$, while $G$ has treewidth $|X|+O(1)$. We rule out algorithms running in time $O^*((r+1-\epsilon)^{k})$ for Deletion to $r$-Colorable parameterized by $k=|X|$. For clique-width, we rule out time $O^*((2^r-\epsilon)^k)$ for Deletion to $r$-Colorable, where $X$ is now allowed to consist of $k$ twinclasses. There are further results on Vertex Cover, Dominating Set and Maximum Cut. All lower bounds are matched by existing and newly designed algorithms.
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