Individual Rationality in Constrained Hedonic Games: Friends, Enemies, and Neutrals

2026-08-14

AI Summary

This study investigates the computational complexity of individually rational coalition structures in hedonic games under size constraints. Through classical and parameterized complexity analyses combined with graph coloring reductions, it reveals that adversarial structures fundamentally drive intractability in friend-oriented models. The research comprehensively characterizes the boundary conditions under which symmetry and size constraints influence complexity, establishing a systematic taxonomy of computational hardness. It identifies the critical structural properties responsible for NP-hardness while delineating tractable cases solvable in polynomial time. These findings provide a rigorous theoretical foundation for algorithm design in constrained cooperative games, clarifying precisely when efficient computation is feasible versus inherently intractable within this important class of hedonic game models.

Abstract

We study constrained coalition formation in games induced by friends, enemies, and neutrals, under the two standard refinements of additively separable preferences: friend-oriented and enemy-oriented. We ask for partitions that are individually rational (IR), while additionally requiring exactly $k$ non-empty coalitions, each satisfying a prescribed lower and upper bound on its size. Although IR alone is trivial to satisfy for any hedonic game, the size constraints make it computationally intractable to decide whether a feasible partition exists. The two models tell strikingly different stories. Under enemy-oriented preferences, the problem collapses to size-constrained graph coloring, and its complexity follows accordingly. Under friend-oriented preferences, however, the picture is far more intricate, and is governed by the enmity structure rather than the friendships. The complexity is further shaped by two factors: how strict the imposed size requirements are, and whether relationships are symmetric or asymmetric, with several cases turning out tractable in the symmetric setting but intractable once asymmetry is allowed. Charting this boundary in terms of both classical and parameterized complexity, we provide a complete understanding of which properties of the friend/enemy structure are responsible for hardness.

Problem

• Constrained Hedonic Games
• Individual Rationality
• Coalition Formation
• Computational Complexity
• Size Constraints

Innovation

• Constrained Hedonic Games
• Individual Rationality
• Parameterized Complexity
• Friend/Enemy-oriented Preferences
• Coalition Size Constraints