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Improved Approximation Ratio for Strategyproof Facility Location on a Cycle

Krzysztof Rogowski, Marcin Dziubiński

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.

Improved Approximation Ratio for Strategyproof Facility Location on a Cycle

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 agents. We show that there exists an SP mechanism that attains an approximation ratio of with respect to the sum of costs of the agents, thus improving the best known upper bound of in the cases of . 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.
Paper Structure (19 sections, 12 theorems, 53 equations, 1 figure)

This paper contains 19 sections, 12 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our contribution
  4. Preliminaries
  5. Common Mechanism Properties
  6. Known SP mechanisms
  7. Analysis
  8. Normalization of profiles
  9. Cut of the profile
  10. Estimations of the social cost after the cut
  11. Estimating $\phi(b)$
  12. Proof of the main theorem
  13. Experiments
  14. Setup
  15. Results
  16. ...and 4 more sections

Key Result

Theorem 1

For any set of agents with an odd cardinality and a cyclic graph $G$ of length $1$, there exists a strategyproof mechanism $M$ whose approximation ratio is bounded from above by $7/4$.

Figures (1)

  • Figure 1: The maximum value of the approximation ratio of the $RD+PCD$ mechanism in profiles with $n$ agents, whose reports are restricted to $G_k$. The dotted line highlights the results calculated with the restriction that the number of different points reported by agents is limited to $3$.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • proof
  • Definition 5
  • ...and 23 more