Discrete Truthful Heterogeneous Two-Facility Location: The Line and Beyond

2026-07-23

AI Summary

This study addresses the design of strategyproof mechanisms for dual-facility location on discrete graphs, where heterogeneous agents must be assigned to one of two facilities they approve, with the objective of minimizing the total social distance cost. For line graphs, the authors propose a novel mechanism combining a fixed parity median rule with a strategyproof local covering scheme, achieving the optimal approximation ratio of 4/3 and thereby closing a long-standing theoretical gap. For general connected graphs, they devise a strategyproof 2-approximation mechanism and establish a lower bound of 3/2 on the approximation ratio for the specific case of $K_{1,3}$ (claw) graphs, revealing an inherent computational hardness tied to this graph structure.

Abstract

We study deterministic strategyproof mechanisms for discrete heterogeneous two-facility location. In our model, $n$ agents occupy distinct nodes of a connected graph and privately report non-empty approval preferences over two facilities, which must be placed at distinct nodes. The cost of an agent is her total distance from the facilities she approves, and the objective is to minimize the social cost (the total cost of the agents). For the line graph, the best possible approximation ratio of deterministic strategyproof mechanisms was previously shown to lie between $4/3$ and $17/4$. We close this gap by designing an optimal $4/3$-approximate mechanism that combines a fixed-parity median rule, which suffices for instances with $n\ge7$, and strategyproof local overrides for the remaining smaller cases. Beyond the line, we design a deterministic strategyproof $2$-approximate mechanism for every connected graph and prove a lower bound of $3/2$ on the graph $K_{1,3}$.

Problem

• truthful mechanism
• heterogeneous facility location
• social cost minimization
• strategyproofness
• approximation ratio

Innovation

• strategyproof mechanism
• two-facility location
• approximation ratio
• discrete preferences
• graph topology