Approximation Algorithms for Edge Partitioned Vertex Cover Problems

TL;DR

This paper introduces a new approximation algorithm for the Partition Vertex Cover problem, utilizing a novel LP relaxation with knapsack cover inequalities, achieving an $O( ext{log } r)$ integrality gap.

Contribution

The paper presents the first LP-based approximation algorithm with matching bounds for the Partition Vertex Cover problem, extending to more general cases.

Findings

LP relaxation with knapsack cover inequalities has an $O( ext{log } r)$ integrality gap.

Matching upper and lower bounds on approximability are established.

The approach generalizes to broader problem settings.

Abstract

We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G = (V,E), a positive cost function c: V-> Z^{+}, a partition $P_1,..., P_r$ of the edge set $E$, and a parameter $k_i$ for each partition $P_i$. The goal is to find a minimum cost set of vertices which cover at least $k_i$ edges from the partition $P_i$. We call this the Partition Vertex Cover problem. In this paper, we give matching upper and lower bound on the approximability of this problem. Our algorithm is based on a novel LP relaxation for this problem. This LP relaxation is obtained by adding knapsack cover inequalities to a natural LP relaxation of the problem. We show that this LP has integrality gap of $O(log r)$, where $r$ is the number of sets in the partition of the edge set. We also extend our result to more general settings.