Hedonic Games and Treewidth Revisited

TL;DR

This paper analyzes the computational complexity of finding Nash stable solutions in Additively Separable Hedonic Games, especially focusing on the influence of graph treewidth and degree, providing new algorithms and complexity bounds.

Contribution

It introduces an improved fixed-parameter algorithm for stability in ASHGs based on treewidth and degree, corrects previous claims, and establishes tight complexity bounds under ETH.

Findings

New algorithm with $( ext{degree} imes ext{treewidth})^{O( ext{degree} imes ext{treewidth})}$ complexity.

Proves no $t^{o(t)}$ dependence algorithm exists unless ETH fails.

Shows Nash Stability is NP-hard on stars, but Connected Nash Stability is solvable in pseudo-polynomial time for fixed treewidth.

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 $\Delta$. The current best FPT algorithm for this case was claimed by Peters [AAAI 2016], with time complexity roughly $2^{O(\Delta^5t)}$. We present an algorithm with parameter dependence $(\Delta t)^{O(\Delta t)}$, significantly improving upon the parameter dependence on $\Delta$ 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 $\Delta$ 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.