Set Covering Problems with General Objective Functions

TL;DR

This paper introduces a generalized set cover problem parameterized by p, unifies several known problems, and provides approximation algorithms with proven bounds, along with complexity results for related graph coloring and generalized problems.

Contribution

It generalizes multiple set cover variants into a single framework, offers a greedy approximation with optimal bounds, and extends analysis to related coloring and concave function minimization problems.

Findings

Greedy algorithm approximates within (p+1)^{1/p} for p>0.

Approximation bounds are tight unless P=NP.

Complexity results for related graph coloring and generalized set cover variants.

Abstract

We introduce a parameterized version of set cover that generalizes several previously studied problems. Given a ground set V and a collection of subsets S_i of V, a feasible solution is a partition of V such that each subset of the partition is included in one of the S_i. The problem involves maximizing the mean subset size of the partition, where the mean is the generalized mean of parameter p, taken over the elements. For p=-1, the problem is equivalent to the classical minimum set cover problem. For p=0, it is equivalent to the minimum entropy set cover problem, introduced by Halperin and Karp. For p=1, the problem includes the maximum-edge clique partition problem as a special case. We prove that the greedy algorithm simultaneously approximates the problem within a factor of (p+1)^1/p for any p in R^+, and that this is the best possible unless P=NP. These results both generalize and simplify previous results for special cases. We also consider the corresponding graph coloring problem, and prove several tractability and inapproximability results. Finally, we consider a further generalization of the set cover problem in which we aim at minimizing the sum of some concave function of the part sizes. As an application, we derive an approximation ratio for a Rent-or-Buy set cover problem.