Randomized algorithm for the partial set multicover problem

In this paper we analyze the minimum cardinality PARTIAL SET b-MULTICOVER problem in uniform and regular hypergraphs. The problem is defined as follows: Let k ϵ ≥1, b ≥ 2 and a hypergraph H = (V,E) with maximum vertex degree Δ and maximum edge size l, a PARTIAL SET b-MULTICOVER in H is a set of edges C ⊆ E such that every vertex in a subset U ⊆ V with |U |≥ k, belongs to at least b edges in C. PARTIAL SET b-MULTICOVER problem is the problem of finding a PARTIAL SET b-MULTICOVER of minimum cardinality. We present a randomized algorithm of hybrid type for this problem, combining LP-based randomized rounding with greedy repairing. We achieve an approximation ratio of α(n,k)n/k(Δ-b+1) with α(n,k) < 2 a factor depends on n and k for hypergraphs with lϵO(n1/5). Furthermore we consider the SET b-MULTICOVER problem in hypergraphs i.e., the PARTIAL SET b-MULTICOVER problem for k = n. It remained an open problem whether an approximation ratio of α(Δ -- b+1) with a constant α > 1 can be proved unless P = NP. This was conjectured by Peleg, Schechtman and Wool (Algorithmica 1997). We present a randomized algorithm for SET b-MULTICOVER, and achieve an approximation ratio of (1 - 1/2√2√n) (Δ -- b+1) for hypergraphs with maximum edge size lϵO(n1/2). The results for both problems presented in this paper improve for large set of instances over the known results.