Tight approximation bounds for combinatorial frugal coverage algorithms

We consider the frugal coverage problem, an interesting variation of set cover defined as follows. Instances of the problem consist of a universe of elements and a collection of sets over these elements; the objective is to compute a subcollection of sets so that the number of elements it covers plus the number of sets not chosen is maximized. The problem was introduced and studied by Huang and Svitkina (Proceedings of the 29th IARCS annual conference on foundations of software technology and theoretical computer science (FSTTCS), pp. 227–238, 2009) due to its connections to the donation center location problem. We prove that the greedy algorithm has approximation ratio at least 0.782, improving a previous bound of 0.731 in Huang and Svitkina (Proceedings of the 29th IARCS annual conference on foundations of software technology and theoretical computer science (FSTTCS), pp. 227–238, 2009). We also present a further improvement that is obtained by adding a simple corrective phase at the end of the execution of the greedy algorithm. The approximation ratio achieved in this way is at least 0.806. Finally, we consider a packing based algorithm that uses semi-local optimization, and show that its approximation ratio is not less than 0.872. Our analysis is based on the use of linear programs which capture the behavior of the algorithms in worst-case examples. The obtained bounds are proved to be tight

Donation Center Location Problem

We introduce and study the donation center location problem, which has an additional application in network testing and may also be of independent interest as a general graphtheoretic problem. Given a set of agents and a set of centers, where agents have preferences over centers and centers have capacities, the goal is to open a subset of centers and to assign a maximum-sized subset of agents to their most-preferred open centers, while respecting the capacity constraints. We prove that in general, the problem is hard to approximate within n 1/2−ɛ for any ɛ> 0. In view of this, we investigate two special cases. In one, every agent has a bounded number of centers on her preference list, and in the other, all preferences are induced by a line-metric. We present constant-factor approximation algorithms for the former and exact polynomial-time algorithms for the latter. Of particular interest among our techniques are an analysis of the greedy algorithm for a variant of the maximum coverage problem called frugal coverage, the use of maximum matching subroutine with subsequent modification, analyzed using a counting argument, and a reduction to the independent set problem on terminal intersection graphs, which we show to be a subclass of trapezoid graphs.

Tight Approximation Bounds for Greedy Frugal Coverage Algorithms

Abstract. We consider the frugal coverage problem, an interesting vari-ation of set cover defined as follows. Instances of the problem consist of a universe of elements and a collection of sets over these elements; the objective is to compute a subcollection of sets so that the number of elements it covers plus the number of sets not chosen is maximized. The problem was introduced and studied by Huang and Svitkina [7] due to its connections to the donation center location problem. We prove that the greedy algorithm has approximation ratio at least 0.782, improving a previous bound of 0.731 in [7]. We also present a further improvement that is obtained by adding a simple corrective phase at the end of the execution of the greedy algorithm. The approximation ratio achieved in this way is at least 0.806. Our analysis is based on the use of linear programs which capture the behavior of the algorithms in worst-case examples. The obtained bounds are proved to be tight.