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Strategyproof Facility Location in Perturbation Stable Instances

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Part of the Lecture Notes in Computer Science book series (LNISA,volume 13112)

Abstract

We study the approximability of k-Facility Location games on the real line by strategyproof mechanisms without payments. To circumvent impossibility results for \(k \ge 3\), we focus on \(\gamma \)-(perturbation) stable instances, where the optimal agent clustering is not affected by moving any subset of consecutive agent locations closer to each other by a factor at most \(\gamma \ge 1\). We show that the optimal solution is strategyproof in \((2+\sqrt{3})\)-stable instances, if it does not include any singleton clusters, and that allocating the facility to the agent next to the rightmost one in each optimal cluster is strategyproof and \((n-2)/2\)-approximate for 5-stable instances (even if singleton clusters are present), where n is the number of agents. On the negative side, we show that for any \(k \ge 3\) and any \(\delta > 0\), deterministic anonymous strategyproof mechanisms suffer an unbounded approximation ratio in \((\sqrt{2}-\delta )\)-stable instances. Moreover, we prove that allocating the facility to a random agent of each optimal cluster is strategyproof and 2-approximate in 5-stable instances.

This work was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant”, project BALSAM, HFRI-FM17-1424. The full version of this work is available at https://arxiv.org/abs/2107.11977.

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Notes

  1. 1.

    A mechanism is anonymous if its outcome depends only on the agent locations, not on their identities.

  2. 2.

    The notion of \(\gamma \)-metric stability is “stricter” than standard \(\gamma \)-stability in the sense that the former excludes some perturbations allowed by the latter. Hence, the class of \(\gamma \)-metric stable instances includes the class of \(\gamma \)-stable instances. More generally, the stricter a notion of stability is, the larger the class of instances qualified as stable, and the more general the positive results that one gets. Similarly, for any \(\gamma ' > \gamma \ge 1\), the class of \(\gamma \)(-metric) stable instances includes the class of \(\gamma '\)(-metric) instances. Hence, a smaller value of \(\gamma \) makes a positive result stronger and more general.

  3. 3.

    E.g., let \(k = 2\) and consider the \(\varTheta (\gamma )\)-stable instance \((0, 1-\varepsilon , 1, 6\gamma , 6\gamma +\varepsilon , 6\gamma +1, 6\gamma +1+\varepsilon , 6\gamma +2)\), for any \(\gamma \ge 1\). Then, the agent at location \(6\gamma \) can decrease its connection cost (from 1) to \(\varepsilon \) by deviating to location \((6\gamma )^2\).

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Fotakis, D., Patsilinakos, P. (2022). Strategyproof Facility Location in Perturbation Stable Instances. In: Feldman, M., Fu, H., Talgam-Cohen, I. (eds) Web and Internet Economics. WINE 2021. Lecture Notes in Computer Science(), vol 13112. Springer, Cham. https://doi.org/10.1007/978-3-030-94676-0_6

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  • DOI: https://doi.org/10.1007/978-3-030-94676-0_6

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