Donation Center Location Problem

has an additional application in network testing and may also be of independent interest as a general graph-theoretic 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 $n^{1/2-\epsilon}$ for any $\epsilon>0$. 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.

This is joint work with Zoya Svitkina