A Primal-Dual Randomized Algorithm for the Online Weighted Set Multi-cover Problem

Given a ground set U of n elements and a family of m subsets S = {S-i : S-i subset of U}. Each subset S is an element of S has a positive cost c(S) and every element e. U is associated with an integer coverage requirement r(e) > 0, which means that e has to be covered at least r(e) times. The weighted set multi-cover problem asks for the minimum cost subcollection which covers all of the elements such that each element e is covered at least r(e) times. In this paper, we study the online version of the weighted set multicover problem. We give a randomized algorithm with competitive ratio 8(1+ lnm) lnn for this problem based on the primal-dual method, which improve previous competitive ratio 12 logmlog n for the online set multicover problem that is the special version where each cost c(S) is 1 for every subset S.