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Strategyproof Facility Location in Perturbation Stable Instances

Dimitris Fotakis, Panagiotis Patsilinakos

TL;DR

This work studies strategyproof facility location on the line under perturbation stability, addressing the gap between impossibility results for general instances and the need for practical incentive-compatible mechanisms. It shows that deterministic mechanisms can achieve bounded or even constant approximation on $5$-stable instances (via the AlmostRightmost approach) and precisely characterizes cases where the optimal solution is SP for $2+\sqrt{3}$-stable instances without singleton clusters. It also proves a strong impossibility: for $k\ge 3$, no deterministic anonymous SP mechanism attains a bounded approximation on $(\sqrt{2}-\delta)$-stable instances, highlighting limits of stability-based tractability. On the randomized side, a simple Random mechanism achieves a $2$-approximation for $5$-stable instances, offering a robust alternative when deterministic guarantees fail. Overall, the paper advances understanding of how stability assumptions can yield nontrivial, implementable SP mechanisms with provable approximation guarantees in a natural clustering setting.

Abstract

We consider $k$-Facility Location games, where $n$ strategic agents report their locations on the real line, and a mechanism maps them to $k\ge 2$ facilities. Each agent seeks to minimize her distance to the nearest facility. We are interested in (deterministic or randomized) strategyproof mechanisms without payments that achieve a reasonable approximation ratio to the optimal social cost of the agents. To circumvent the inapproximability of $k$-Facility Location by deterministic strategyproof mechanisms, we restrict our attention to perturbation stable instances. An instance of $k$-Facility Location on the line is $γ$-perturbation stable (or simply, $γ$-stable), for some $γ\ge 1$, if the optimal agent clustering is not affected by moving any subset of consecutive agent locations closer to each other by a factor at most $γ$. We show that the optimal solution is strategyproof in $(2+\sqrt{3})$-stable instances whose optimal solution does not include any singleton clusters, and that allocating the facility to the agent next to the rightmost one in each optimal cluster (or to the unique agent, for singleton clusters) is strategyproof and $(n-2)/2$-approximate for $5$-stable instances (even if their optimal solution includes singleton clusters). On the negative side, we show that for any $k\ge 3$ and any $δ> 0$, there is no deterministic anonymous mechanism that achieves a bounded approximation ratio and is strategyproof in $(\sqrt{2}-δ)$-stable instances. We also prove that allocating the facility to a random agent of each optimal cluster is strategyproof and $2$-approximate in $5$-stable instances. To the best of our knowledge, this is the first time that the existence of deterministic (resp. randomized) strategyproof mechanisms with a bounded (resp. constant) approximation ratio is shown for a large and natural class of $k$-Facility Location instances.

Strategyproof Facility Location in Perturbation Stable Instances

TL;DR

This work studies strategyproof facility location on the line under perturbation stability, addressing the gap between impossibility results for general instances and the need for practical incentive-compatible mechanisms. It shows that deterministic mechanisms can achieve bounded or even constant approximation on -stable instances (via the AlmostRightmost approach) and precisely characterizes cases where the optimal solution is SP for -stable instances without singleton clusters. It also proves a strong impossibility: for , no deterministic anonymous SP mechanism attains a bounded approximation on -stable instances, highlighting limits of stability-based tractability. On the randomized side, a simple Random mechanism achieves a -approximation for -stable instances, offering a robust alternative when deterministic guarantees fail. Overall, the paper advances understanding of how stability assumptions can yield nontrivial, implementable SP mechanisms with provable approximation guarantees in a natural clustering setting.

Abstract

We consider -Facility Location games, where strategic agents report their locations on the real line, and a mechanism maps them to facilities. Each agent seeks to minimize her distance to the nearest facility. We are interested in (deterministic or randomized) strategyproof mechanisms without payments that achieve a reasonable approximation ratio to the optimal social cost of the agents. To circumvent the inapproximability of -Facility Location by deterministic strategyproof mechanisms, we restrict our attention to perturbation stable instances. An instance of -Facility Location on the line is -perturbation stable (or simply, -stable), for some , if the optimal agent clustering is not affected by moving any subset of consecutive agent locations closer to each other by a factor at most . We show that the optimal solution is strategyproof in -stable instances whose optimal solution does not include any singleton clusters, and that allocating the facility to the agent next to the rightmost one in each optimal cluster (or to the unique agent, for singleton clusters) is strategyproof and -approximate for -stable instances (even if their optimal solution includes singleton clusters). On the negative side, we show that for any and any , there is no deterministic anonymous mechanism that achieves a bounded approximation ratio and is strategyproof in -stable instances. We also prove that allocating the facility to a random agent of each optimal cluster is strategyproof and -approximate in -stable instances. To the best of our knowledge, this is the first time that the existence of deterministic (resp. randomized) strategyproof mechanisms with a bounded (resp. constant) approximation ratio is shown for a large and natural class of -Facility Location instances.

Paper Structure

This paper contains 17 sections, 16 theorems, 33 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Other Related Work
  3. Notation, Definitions and Preliminaries
  4. Perturbation Stability on the Line: Definition and Properties
  5. A Deterministic Mechanism Resistant to Singleton Deviations
  6. Low Stability and Inapproximability by Deterministic Mechanisms
  7. Image Sets, Holes and Well-Separated Instances
  8. The Proof of the Impossibility Result
  9. A Randomized Mechanism with Constant Approximation
  10. The Proof of Lemma \ref{['l:separated-clusters']}
  11. The Proof of Lemma \ref{['3_stab_lemma']}
  12. The Proof of Lemma \ref{['l:singleton_deviation_5']}
  13. Proofs of Auxiliary Lemmas Used in the Proof of Theorem \ref{['thm:lower_bound']}
  14. The Proof of Lemma \ref{['l:nice-allocation']}
  15. The Proof of Lemma \ref{['l:movingRight']}
  16. ...and 2 more sections

Key Result

proposition thmcounterproposition

Let $\gamma \geq 2$ and let $\hbox{\boldmath$x$}$ be any $\gamma$-stable instance, with unique optimal clustering $C_1, \ldots, C_k$ and optimal centers $c_1, \ldots, c_k$. Then, for all clusters $C_i$ and $C_j$, with $i \neq j$, and all locations $x \in C_i$ and $y \in C_j$, $d(x, y) > (\gamma - 1)

Figures (1)

  • Figure 1: An example of a so-called singleton deviation. The deviating agent (grey) declares a remote location, becomes a singleton cluster, and essentially turns the remaining agents into a $(k-1)$-Facility Location instance. Thus, the deviating agent can benefit from her singleton deviation, due to the subsequent cluster merge.

Theorems & Definitions (36)

  • definition thmcounterdefinition: $\gamma$-Pertrubation and $\gamma$-Stability
  • proposition thmcounterproposition
  • lemma thmcounterlemma: Cluster-Separation Property
  • corollary thmcountercorollary
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof : Sketch.
  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma: Proposition 2.2, FT12
  • ...and 26 more