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) with a maximum edge size f, a cost function w : V → Z⁺, and edge subsets P₁, P₂, . . ., P_r of E along with covering requirements k₁, k₂, . . ., 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 a generalization of the edge-partitioned vertex cover problem considered by Bera et al. We present a primal-dual algorithm yielding an (f · H_r + H_r)-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 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.