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

2025-07-22

AI Summary

This paper studies multi-objective optimization for $k$-facility location (or $k$-committee selection) in metric spaces, focusing on the compatibility between individual client objectives—minimizing distance to the nearest facility (sum or max)—and global system objectives—minimizing the sum or maximum of all client distances. We conduct the first systematic analysis of approximate Pareto feasibility across the four mixed objective pairs: sum-sum, sum-max, max-sum, and max-max. Leveraging constructive approximation algorithms and structural properties of metric spaces, we prove that for any pair of objectives, there exists a $k$-subset achieving constant-factor approximations to both respective optima simultaneously. This demonstrates that these competing objectives need not be traded off against one another. Our result establishes a theoretical foundation and algorithmic guarantee for designing fair, robust, and efficient facility locations and representative committees.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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 $iin C$ 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.

Problem

• Study compatibility of multiple objectives in facility location
• Analyze trade-offs between sum and max distance metrics
• Find solutions optimal for multiple objectives simultaneously

Innovation

• Simultaneously optimize multiple facility location objectives
• Balance sum and maximum distance metrics
• Provide close-to-optimum solutions for paired objectives