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arXiv 1407.2033cs.DS
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加权FPT算法的一个多元框架

A Multivariate Framework for Weighted FPT Algorithms

Hadas Shachnai, Meirav Zehavi

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摘要：

我们介绍了一种新的多元方法来解决加权参数化问题。在我们的模型中，给定一个最小化（最大化）问题的大小$n$的实例，以及一个参数$W \geq 1$，我们寻求权重最多（或至少）$W$的解。我们使用我们的一般框架，以获得有效的算法等基本图的问题，顶点覆盖，3-命中集，边支配集和最大内部出分支。这些问题的最著名的算法承认运行时间的形式$c^W n^{O(1)}$，对于一些常数$c>1$。我们将这些运行时间改进为$c^s n^{O(1)}$，其中$s\leq W$是最多（至少）$W$的解的最小大小。如果不存在这样的解，则为$s=\min\{W,m\}$，其中$m$是解的最大大小。显然，$s$可以显著小于$W$。特别是，我们的算法的运行时间是（几乎）相同的最知名的$O^*$运行时间的未加权的变种。因此，我们在$1.381^s n^{O(1)}$时间和$n^{O(1)}$空间中求解 * 顶点覆盖的加权版本。* 3-在$2.168^s n^{O(1)}$时间和$n^{O(1)}$空间中的命中集。* $2.315^s n^{O(1)}$时间和$n^{O(1)}$空间中的边支配集。* $6.855^s n^{O(1)}$时间和空间中的最大内部外分支。我们进一步表明，加权顶点覆盖和加权边支配集承认快速算法，其运行时间的形式是$c^t n^{O(1)}$，其中$t \leq s$是一个解决方案的最小尺寸。

英文摘要：

We introduce a novel multivariate approach for solving weighted parameterized problems. In our model, given an instance of size $n$ of a minimization (maximization) problem, and a parameter $W \geq 1$, we seek a solution of weight at most (or at least) $W$. We use our general framework to obtain efficient algorithms for such fundamental graph problems as Vertex Cover, 3-Hitting Set, Edge Dominating Set and Max Internal Out-Branching. The best known algorithms for these problems admit running times of the form $c^W n^{O(1)}$, for some constant $c>1$. We improve these running times to $c^s n^{O(1)}$, where $s\leq W$ is the minimum size of a solution of weight at most (at least) $W$. If no such solution exists, $s=\min\{W,m\}$, where $m$ is the maximum size of a solution. Clearly, $s$ can be substantially smaller than $W$. In particular, the running times of our algorithms are (almost) the same as the best known $O^*$ running times for the unweighted variants. Thus, we solve the weighted versions of * Vertex Cover in $1.381^s n^{O(1)}$ time and $n^{O(1)}$ space. * 3-Hitting Set in $2.168^s n^{O(1)}$ time and $n^{O(1)}$ space. * Edge Dominating Set in $2.315^s n^{O(1)}$ time and $n^{O(1)}$ space. * Max Internal Out-Branching in $6.855^s n^{O(1)}$ time and space. We further show that Weighted Vertex Cover and Weighted Edge Dominating Set admit fast algorithms whose running times are of the form $c^t n^{O(1)}$, where $t \leq s$ is the minimum size of a solution.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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