Facility location problems: multiple objectives and strategic clients

Abstract

We study a version of the metric facility location problem (or, equivalently, variants of the committee selection problem) in which we must choose k facilities in an arbitrary metric space to serve some set of clients C. We consider the case where the exact locations of the clients are known as well as the case where clients are strategic. For the first case, unlike most previous work, we do not focus on a single objective to optimize (e.g., the total distance from clients to the facility, or the maximum distance, etc.), but instead attempt to optimize several different objectives simultaneously. The first part of the thesis considers the classic single-facility location problem. We consider the l-centrum family of objectives, which includes the total distance, max distance, and many others. We present tight bounds on how well any pair of such objectives (e.g., max and sum) can be simultaneously approximated compared to their optimum outcomes. In particular, we show that for any such pair of objectives, it is always possible to choose an outcome which simultaneously approximates both objectives within a factor of 1 + √2, and give a precise characterization of how this factor improves as the two objectives being optimized become more similar. For q > 2 different centrum objectives, we show that it is always possible to approximate all q of these objectives within a small constant, and that this constant approaches 3 as q → ∞. Our results show that when optimizing only a few simultaneous objectives, it is always possible to form an outcome which is a significantly better than 3 approximation for all of these objectives. The second part of the thesis extends this perspective to the multi-facility setting. We consider four different objectives, where each client i ∈ C attempts to minimize either the sum or the maximum of its distance to the chosen facilities, and where the overall objective either considers the sum or the maximum of the individual client costs, in contrast to the classical setting where each client is assigned to their closest facility. We study how compatible these objectives are with each other, and show the existence of solutions which are simultaneously close-to-optimum for any pair of the above objectives. Our results show that when choosing a set of facilities or a representative committee, it is often possible to form a solution which is good for several objectives at the same time, instead of sacrificing one desideratum to achieve another. The last part of the thesis considers the second case, where clients are strategic. We study Nash equilibria in strategic facility location games where clients are located in an arbitrary metric space. Specifically, there are n clients, and the goal is to choose a facility from a set of given locations, so that the total distance from the clients to the facility is as small as possible. While some of the clients are always truthful, λ of them are strategic, and will lie about their location if it benefits them. We quantify how the fraction of strategic clients affects the existence and quality of Nash equilibrium and strong equilibrium solutions, and note that even for relatively large λ, the properties of these solutions can be much betterthan the results of fully strategyproof mechanisms. For Nash equilibrium, we show that it always exists, and the price of stability is very close to 1. More importantly, we prove that all Nash equilibria are within a factor of at most (n+2λ)/(n−2λ) from the optimum solution, and that this price of anarchy bound is almost tight. While strong equilibrium may not exist for this setting, we prove that it always exists for line metrics, and its cost is at most (n+λ)/(n−λ) times that of the optimum.