A Randomised Approximation Algorithm for the Hitting Set Problem

Abstract

Let \(\mathcal{H}=(V,\mathcal{E})\) be a hypergraph with vertex set V and edge set \(\mathcal{E} \), where n : = |V| and \(m := |{\cal E}|\). Let l be the maximum size of an edge and Δ be the maximum vertex degree. A hitting set (or vertex cover) in \(\mathcal{H}\) is a set of vertices from V in which all edges are incident. The hitting set problem is to find a hitting set of minimum cardinality. It is known that an approximation ratio of l can be achieved easily. On the other side, for constant l, an approximation ratio better than l cannot be achieved in polynomial time under the unique games conjecture (Khot and Ragev 2008). Thus breaking the l-barrier for significant classes of hypergraphs is a complexity-theoretic and algorithmically interesting problem, which has been studied by several authors (Krivelevich (1997), Halperin (2000), Okun (2005)). We propose a randomised algorithm of hybrid type for the hitting set problem, which combines LP-based randomised rounding, graphs sparsening and greedy repairing and analyse it in different environments. For hypergraphs with \(\Delta = O(n^{\frac14})\) and \(l=O(\sqrt{n})\) we achieve an approximation ratio of \(l\left(1-\frac{c}{\Delta}\right)\), for some constant c > 0, with constant probability. In the case of l-uniform hypergraphs, l and Δ being constants, we prove by analysing the expected size of the hitting set and using concentration inequalities, a ratio of \(l\left(1-\frac{l-1}{4\Delta}\right)\). Moreover, for quasi-regularisable hypergraphs, we achieve an approximation ratio of \(l\left(1-\frac{n}{8m}\right)\). We show how and when our results improve over the results of Krivelevich, Halperin and Okun.