Optimal dynamic program for r-domination problems over tree decompositions

G. Borradaile, Hung Le
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引用次数: 29

Abstract

There has been recent progress in showing that the exponential dependence on treewidth in dynamic programming algorithms for solving NP-hard problems are optimal under the Strong Exponential Time Hypothesis (SETH). We extend this work to $r$-domination problems. In $r$-dominating set, one wished to find a minimum subset $S$ of vertices such that every vertex of $G$ is within $r$ hops of some vertex in $S$. In connected $r$-dominating set, one additionally requires that the set induces a connected subgraph of $G$. We give a $O((2r+1)^{\mathrm{tw}} n)$ time algorithm for $r$-dominating set and a $O((2r+2)^{\mathrm{tw}} n^{O(1)})$ time algorithm for connected $r$-dominating set in $n$-vertex graphs of treewidth $\mathrm{tw}$. We show that the running time dependence on $r$ and $\mathrm{tw}$ is the best possible under SETH. This adds to earlier observations that a "+1" in the denominator is required for connectivity constraints.
树分解上r控制问题的最优动态规划
在强指数时间假设(SETH)下,求解np困难问题的动态规划算法对树宽度的指数依赖性是最优的。我们将这一工作推广到$r$支配问题。在支配集$r$中,我们希望找到一个由顶点组成的最小子集$S$,使得$G$的每个顶点都在$S$中某个顶点的$r$跳以内。在连通的$r$支配集中,人们还需要该集引出$G$的连通子图。我们给出了$r$支配集$O((2r+1)^{\ mathm {tw}} n)$ time算法和$O((2r+2)^{\ mathm {tw}} n^{O(1)})$ time算法,用于$r$支配集在$n$-顶点图$\ mathm {tw}$上的连通$r$支配集。我们证明了在SETH下,对$r$和$\ mathm {tw}$的运行时间依赖性是最好的。这增加了先前的观察,即分母中的“+1”是连接约束所必需的。
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