Metric Facility Assignment with Partial Information

2026-06-04

AI Summary

This work addresses the facility assignment problem in linear metric spaces where only partial metric information is available, aiming to approximately minimize social cost—the sum of agents’ distances to their assigned facilities. It presents the first systematic analysis of how combinations of ordinal preferences (ORD), approval preferences (APP), and inter-facility distances (DIST) affect distortion. The study breaks the well-known lower bound of 3 on distortion achievable with ORD alone, proving that APP combined with DIST yields a tight distortion bound of $1+\sqrt{2}$ under general metrics, and that integrating all three information types achieves a tight bound of 2. Through carefully designed deterministic algorithms and rigorous distortion analysis, the paper establishes tight upper and lower bounds for various information regimes, significantly improving upon classical ORD-only results.

Abstract

We study an assignment problem where a set of agents and a set of facilities lie on a line metric. The goal is to compute an assignment of agents to facilities to approximately minimize the social cost (the total distance of agents from their assigned facilities) given only partial information regarding the metric. Unlike previous work which focused solely on algorithms with access to the ordinal preferences of the agents over the facilities (ORD), we also consider the value of information regarding approval preferences (APP), and inter-facility distances (DIST). For different combinations of these three information types, we establish tight bounds on the distortion of deterministic algorithms, showing that it is possible to improve over the optimal bound of $3$ that can be achieved using only ORD information. Among other results, we show a tight bound of $1+\sqrt{2}$ for APP+DIST which holds even for general metrics, and a tight bound of $2$ for ORD+APP+DIST.

Problem

• assignment problem
• partial information
• social cost
• distortion
• metric space

Innovation

• distortion
• facility assignment
• partial information
• ordinal preferences
• approval preferences