Compatibility of Max and Sum Objectives for Committee Selection and $k$-Facility Location

TL;DR

This paper explores the compatibility of different objectives in the metric facility location and committee selection problems, demonstrating that solutions can often be nearly optimal across multiple criteria simultaneously.

Contribution

It introduces a study of the compatibility between sum and max objectives in facility location and committee selection, showing solutions can be close to optimal for multiple objectives at once.

Findings

Solutions can be simultaneously close-to-optimal for different objectives.

Compatibility of sum and max objectives allows flexible solution design.

Multi-objective solutions reduce trade-offs in facility and committee selection.

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 four different objectives, where each client $i\in C$ 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. Rather than optimizing a single objective at a time, 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.