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Core Stability in Additively Separable Hedonic Games of Low Treewidth

Authors Tesshu Hanaka , Noleen Köhler , Michael Lampis

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Author Details

Tesshu Hanaka
  • Kyushu University, Fukuoka, Japan
Noleen Köhler
  • University of Leeds, UK
Michael Lampis
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France

Funding

  • Hanaka, Tesshu: Partially supported by JSPS KAKENHI Grant Numbers JP21H05852, JP21K17707, JP22H00513, JP23H04388.
  • Köhler, Noleen: Partially supported by ANR project ANR-18-CE40-0025-01 (ASSK).
  • Lampis, Michael: Supported by ANR project ANR-21-CE48-0022 (S-EX-AP-PE-AL).

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Tesshu Hanaka, Noleen Köhler, and Michael Lampis. Core Stability in Additively Separable Hedonic Games of Low Treewidth. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 39:1-39:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.39

Abstract

Additively Separable Hedonic Games (ASHGs) are coalition-formation games where we are given a directed graph whose vertices represent n selfish agents and the weight of each arc uv denotes the preferences from u to v. We revisit the computational complexity of the well-known notion of core stability of symmetric ASHGs, where the goal is to construct a partition of the agents into coalitions such that no group of agents would prefer to diverge from the given partition and form a new coalition. For Core Stability Verification (CSV), we first show the following hardness results: CSV remains coNP-complete on graphs of vertex cover 2; CSV is coW[1]-hard parameterized by vertex integrity when edge weights are polynomially bounded; and CSV is coW[1]-hard parameterized by tree-depth even if all weights are from {-1,1}. We complement these results with essentially matching algorithms and color{black}{an FPT algorithm parameterized by the treewidth tw plus the maximum degree Δ (improving a previous algorithm’s dependence from 2^O(twΔ²)} to 2^O(twΔ)).} We then move on to study Core Stability (CS), which one would naturally expect to be even harder than CSV. We confirm this intuition by showing that CS is Σ₂^p-complete even on graphs of bounded vertex cover number. On the positive side, we present a 2^{2^O(Δtw)}n^O(1)-time algorithm parameterized by tw+Δ, which is essentially optimal assuming Exponential Time Hypothesis (ETH). Finally, we consider the notion of k-core stability: k denotes the maximum size of the allowed blocking (diverging) coalitions. We show that k-CSV is coW[1]-hard parameterized by k (even on unweighted graphs), while k-CS is NP-complete for all k ≥ 3 (even on graphs of bounded degree with bounded edge weights).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Hedonic games
  • Treewidth
  • Core stability

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