Improved Approximation Ratio for Strategyproof Facility Location on a Cycle

TL;DR

This paper introduces a new strategyproof mechanism for facility location on a cycle that improves the approximation ratio from 2-2/n to 7/4, using a cycle-cutting technique to relate the problem to a line.

Contribution

It presents a novel SP mechanism with a better approximation ratio for cycle facility location, combining existing mechanisms via randomization and a cycle-cutting approach.

Findings

Achieves an approximation ratio of 7/4 for the problem.

Improves upon the previous bound of 2-2/n for n ≥ 5.

Introduces a cycle-cutting technique to analyze the problem.

Abstract

We study the problem of design of strategyproof in expectation (SP) mechanisms for facility location on a cycle, with the objective of minimizing the sum of costs of $n$ agents. We show that there exists an SP mechanism that attains an approximation ratio of $7/4$ with respect to the sum of costs of the agents, thus improving the best known upper bound of $2-2/n$ in the cases of $n \geq 5$. The mechanism obtaining the bound randomizes between two mechanisms known in the literature: the Random Dictator (RD) and the Proportional Circle Distance (PCD) mechanism of Meir (arXiv:1902.08070). To prove the result, we propose a cycle-cutting technique that allows for estimating the problem on a cycle by a problem on a line.