Approximation algorithm for prize-collecting weighted set cover with fairness constraints

Abstract

Fairness has become one of the hottest concerns in recent research. This paper introduces the prize-collecting weighted set cover problem with fairness constraint (FPCWSC). It is a variant of the minimum weight set cover problem, in which every uncovered element incurs a penalty and the elements are divided into several groups, each group having a minimum number of elements required to be covered. The goal is to minimize the cost of selected sets plus the penalties on those uncovered elements, subject to the constraint that every group has its coverage requirement satisfied. We propose a four-phase algorithm using deterministic rounding twice, followed by a randomized rounding method and a greedy method. In polynomial time, the algorithm computes a feasible solution with an expected approximation ratio $O(f+ln\Delta )$, where $f$ is the maximum number of sets containing a common element and $\Delta$ is the maximum number of groups having nonempty intersection with a set.