A Parameterized Approximation Scheme for Generalized Partial Vertex Cover

Abstract

Partial Vertex Cover is a well-studied generalization of the classic Vertex Cover problem, where we are given a graph $G=(V,E)$ along with a non-negative integer k, and the goal is to cover the maximum number of edges possible by picking exactly k vertices. In this paper, we study a natural extension of Partial Vertex Cover to multiple color classes of the edges. In our problem, we are additionally given a partition of E into m color classes $E_1,E_2,...,E_m$ and coverage requirements $c_i\ge 1$ for all $1\le i\le m$. The goal is to find a subset of vertices of size k that covers at least $\beta \cdot c_i$ edges from each $E_i$ and the contraction factor $\beta \le 1$ is maximized.

As we prove in our paper, the multi-colored extension becomes very difficult to approximate in polynomial time to any reasonable factor. Consequently, we study the parameterized complexity of approximating this problem in terms of various parameters such as k and m. Our main result is a $(1-\epsilon )$-approximation for the problem that runs in time $f(k,m,\epsilon )\cdot \mathop {\hbox {poly}}\nolimits (|V|)$ for some computable function f. As we argue, our result is tight, in the sense that it is not possible to remove the dependence on k or m from the running time of such a $(1-\epsilon )$-approximation.