A Multivariate Framework for Weighted FPT Algorithms
Abstract
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, 3Hitting Set, Edge Dominating Set and Max Internal OutBranching. 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. * 3Hitting 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 OutBranching 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 arXiv:
 arXiv:1407.2033
 Bibcode:
 2014arXiv1407.2033S
 Keywords:

 Computer Science  Data Structures and Algorithms