Approximation Algorithm for Vertex Cover with Multiple Covering Constraints

Abstract

We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph $G=(V,E)$ with a maximum edge size f, a cost function $w: V\rightarrow {\mathbb {Z}}^+$, and edge subsets $P_1,P_2,\ldots ,P_r$ of E along with covering requirements $k_1,k_2,\ldots ,k_r$ for each subset. The objective is to find a minimum cost subset S of V such that, for each edge subset $P_i$, at least $k_i$ edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and the edge-partitioned vertex cover problem considered by Bera et al. We present a primal-dual algorithm yielding an $\left( f \cdot H_r + H_r\right)$-approximation for this problem, where $H_r$ is the $r^{th}$ harmonic number. This improves over the previous ratio of $(3cf\log r)$, where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to the previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality.