Randomized approximation of bounded multicovering problems

Abstract

The problem of finding approximate solutions for a subclass of multicovering problems denoted byILP(k, b) is considered. The problem involves findingxź{0,1}n that minimizes źjxj subject to the constraintAxźb, whereA is a 0---1m×n matrix with at mostk ones per row,b is an integer vector, andb is the smallest entry inb. This subclass includes, for example, the Bounded Set Cover problem whenb=1, and the Vertex Cover problem whenk=2 andb=1. An approximation ratio ofkźb+1 is achievable by known deterministic algorithms. A new randomized approximation algorithm is presented, with an approximation ratio of (kźb+1)(1ź(c/m)1/(kźb+1)) for a small constantc>0. The analysis of this algorithm relies on the use of a new bound on the sum of independent Bernoulli random variables, that is of interest in its own right.