Min-Sum Set Cover on Parallel Machines

Abstract

Consider the classical Min-Sum Set Cover problem: We are given a universe $\mathcal{U}$ of $n$ elements and a collection $\mathcal{S}$ of $k$ subsets of $\mathcal{U}$. The goal is to find a subsequence of sets from $\mathcal{S}$ which covers all elements in $\mathcal{U}$, such that the sum of the covering times of the elements is minimized. This problem can be seen as a scheduling problem on a single machine, where each job represents a set and elements are represented by some kind of utility that is required to be provided by at least one of the jobs. The goal is to schedule the jobs in such a way to minimize the sum of provision times of the utilities. In this paper we consider a natural generalization of this problem to the case of $m$ machines, processing the jobs in parallel. We call this problem Parallel Min-Sum Set Cover. To obtain approximation algorithms for various variants of this task, we exploit a crucial sub-problem called Parallel Densest Subfamily, where the goal is to find an asignment of sets to the machines that maximizes the ratio of the number of covered elements to the length of the assignment. We prove that an $α$-approximation algorithm for this problem implies a $4α$-approximation algorithm for Parallel Min-Sum Set Cover. Then, we show how to find such an assignment using the well known Maximum Coverage Multiple Knapsack problem. In particular, this yields a $\frac{4e}{e-1}+ε$-approximation for identical machines and an $\frac{8e}{e-1}+ε$-approximation for unrelated machines. If the sets are subject to precedence constraints we give a greedy algorithm for unit cost sets, with an $O(k^{2/3})$ approximation ratio and an $O(\log k)$-approximation algorithm for out-forest precedence constraints and identical machines. The latter algorithm uses a reduction to the Group Steiner Orienteering problem which is of independent interest.