A randomised approximation algorithm for the hitting set problem
2014 (English)In: Theoretical Computer Science, ISSN 0304-3975, Vol. 555, 23-34 p.Article in journal (Refereed) Published
Let H = (V, epsilon) be a hypergraph with vertex set V and edge set epsilon, where n := vertical bar V vertical bar and m := vertical bar epsilon vertical bar. Let l be the maximum size of an edge and Delta be the maximum vertex degree. A hitting set (or vertex cover) in H is a subset of 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 hand, for constant l, an approximation ratio better than l cannot be achieved in polynomial time under the unique games conjecture (Khot and Regev, 2008 ). Thus breaking the l-barrier for significant classes of hypergraphs is a complexity-theoretically 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 for different classes of hypergraphs. For hypergraphs with Delta = O(n1/4) and l = O (root n) we achieve an approximation ratio of l(1 - c/Delta), for some constant c greater than 0, with constant probability. For the case of hypergraphs where l and Delta are constants, we prove a ratio of l(1 - l-1/8 Delta). The latter is done by analysing the expected size of the hitting set and using concentration inequalities. Moreover, for quasi-regularisable hypergraphs, we achieve an approximation ratio of l(1 - n/8m). We show how and when our results improve over the results of Krivelevich, Halperin and Okun.
Place, publisher, year, edition, pages
Elsevier , 2014. Vol. 555, 23-34 p.
Approximation algorithms; Probabilistic methods; Randomised rounding; Hitting set; Vertex cover; Greedy algorithms
Computer and Information Science Mathematics
IdentifiersURN: urn:nbn:se:liu:diva-112302DOI: 10.1016/j.tcs.2014.03.029ISI: 000343627600004OAI: oai:DiVA.org:liu-112302DiVA: diva2:765671