Core Stability in Additively Separable Hedonic Games of Low Treewidth

TL;DR

This work analyzes the core stability of Additively Separable Hedonic Games on edge-weighted graphs through structural graph parameters. It identifies a broad spectrum of hardness results for core stability verification and core stability, including Sigma_2^p-completeness on graphs with bounded vertex cover and coW[1]-hardness under several parameters, while also giving targeted algorithmic results: polynomial-time solvability on trees, XP and FPT results for various parameterizations, and a double-exponential-in-treewidth algorithm with ETH-based lower bounds for CS. The paper highlights the intricate interplay between edge weights and graph structure in driving complexity and outlines strong barriers to efficient algorithms, alongside precise positive FPT results in constrained settings. It also extends the investigation to k-core stability, showing limited tractability improvements under fixed k, and proposes directions for future research in more nuanced parameterizations and hedonic game variants.

Abstract

Additively Separable Hedonic Game (ASHG) are coalition-formation games where we are given a graph whose vertices represent $n$ selfish agents and the weight of each edge $uv$ denotes how much agent $u$ gains (or loses) when she is placed in the same coalition as agent $v$. We revisit the computational complexity of the well-known notion of core stability of 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 (blocking) coalition. Since both finding a core stable partition and verifying that a given partition is core stable are intractable problems ($Σ_2^p$-complete and coNP-complete respectively) we study their complexity from the point of view of structural parameterized complexity, using standard graph-theoretic parameters, such as treewidth.

Figures (2)

• Figure 1: The graph $(G,w)$ constructed in the proof of \ref{['thm:csv:FPT:vc']}. The vertex cover is marked by blue squares. The initial partition is given by dot-dashed boxes, sngletons are omitted.
• Figure 5: The graph $\widehat{G}$ constructed in the proof of Theorem \ref{['thm:3csf:3col']} for an instance of $3$-colouring $G$ consisting of a single edge. In the illustration edges of weight $\rho$ are omitted. The core stable partition indicated by dot-dashed boxes corresponds to coloring the two vertices of $G$ with two different colors.

Theorems & Definitions (19)

• Definition 1: Core stability
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Lemma 10