Hedonic Games and Treewidth Revisited

Tesshu Hanaka, Michael Lampis

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We revisit the complexity of the well-studied notion of Additively Separable Hedonic Games (ASHGs). Such games model a basic clustering or coalition formation scenario in which selfish agents are represented by the vertices of an edge-weighted digraph G = (V,E), and the weight of an arc uv denotes the utility u gains by being in the same coalition as v. We focus on (arguably) the most basic stability question about such a game: Given a graph, does a Nash stable solution exist and can we find it efficiently? We study the (parameterized) complexity of ASHG stability when the underlying graph has treewidth t and maximum degree Δ. The current best FPT algorithm for this case was claimed by Peters [AAAI 2016], with time complexity roughly 2O(Δ5t). We present an algorithm with parameter dependence (Δt)O(Δt), significantly improving upon the parameter dependence on Δ given by Peters, albeit with a slightly worse dependence on t. Our main result is that this slight performance deterioration with respect to t is actually completely justified: we observe that the previously claimed algorithm is incorrect, and that in fact no algorithm can achieve dependence to(t) for bounded-degree graphs, unless the ETH fails. This, together with corresponding bounds we provide on the dependence on Δ and the joint parameter establishes that our algorithm is essentially optimal for both parameters, under the ETH. We then revisit the parameterization by treewidth alone and resolve a question also posed by Peters by showing that Nash Stability remains strongly NP-hard on stars under additive preferences. Nevertheless, we also discover an island of mild tractability: we show that Connected Nash Stability is solvable in pseudo-polynomial time for constant t, though with an XP dependence on t which, as we establish, cannot be avoided.

Original languageEnglish
Title of host publication30th Annual European Symposium on Algorithms, ESA 2022
EditorsShiri Chechik, Gonzalo Navarro, Eva Rotenberg, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772471
DOIs
Publication statusPublished - Sep 1 2022
Event30th Annual European Symposium on Algorithms, ESA 2022 - Berlin/Potsdam, Germany
Duration: Sep 5 2022Sep 9 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume244
ISSN (Print)1868-8969

Conference

Conference30th Annual European Symposium on Algorithms, ESA 2022
Country/TerritoryGermany
CityBerlin/Potsdam
Period9/5/229/9/22

All Science Journal Classification (ASJC) codes

  • Software

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