License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2022.64
URN: urn:nbn:de:0030-drops-170025
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Hanaka, Tesshu ; Lampis, Michael

Hedonic Games and Treewidth Revisited

LIPIcs-ESA-2022-64.pdf (0.8 MB)


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 2^{O(Δ⁵t)}. We present an algorithm with parameter dependence (Δ t)^{O(Δ t)}, 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 t^{o(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.

BibTeX - Entry

  author =	{Hanaka, Tesshu and Lampis, Michael},
  title =	{{Hedonic Games and Treewidth Revisited}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{64:1--64:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-170025},
  doi =		{10.4230/LIPIcs.ESA.2022.64},
  annote =	{Keywords: Hedonic Games, Nash Equilibrium, Treewidth}

Keywords: Hedonic Games, Nash Equilibrium, Treewidth
Collection: 30th Annual European Symposium on Algorithms (ESA 2022)
Issue Date: 2022
Date of publication: 01.09.2022

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