Approximation Algorithm for Vertex Cover with Multiple Covering Constraints

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