Simple Causes of Complexity in Hedonic Games

Hedonic games provide a natural model of coalition formation among self-interested agents. The associated problem of finding stable outcomes in such games has been extensively studied. In this paper, we identify simple conditions on expressivity of hedonic games that are sufficient for the problem of checking whether a given game admits a stable outcome to be computationally hard. Somewhat surprisingly, these conditions are very mild and intuitive. Our results apply to a wide range of stability concepts (core stability, individual stability, Nash stability, etc.) and to many known formalisms for hedonic games (additively separable games, games with W-preferences, fractional hedonic games, etc.), and unify and extend known results for these formalisms. They also have broader applicability: for several classes of hedonic games whose computational complexity has not been explored in prior work, we show that our framework immediately implies a number of hardness results for them.Comment: 7+9 pages, long version of a paper in IJCAI 201

Precise Complexity of the Core in Dichotomous and Additive Hedonic Games

Hedonic games provide a general model of coalition formation, in which a set of agents is partitioned into coalitions, with each agent having preferences over which other players are in her coalition. We prove that with additively separable preferences, it is $\Sigma_2^p$-complete to decide whether a core- or strict-core-stable partition exists, extending a result of Woeginger (2013). Our result holds even if valuations are symmetric and non-zero only for a constant number of other agents. We also establish $\Sigma_2^p$-completeness of deciding non-emptiness of the strict core for hedonic games with dichotomous preferences. Such results establish that the core is much less tractable than solution concepts such as individual stability.Comment: ADT-2017, 15 pages in LNCS styl

Hedonic Games with Graph-restricted Communication

We study hedonic coalition formation games in which cooperation among the players is restricted by a graph structure: a subset of players can form a coalition if and only if they are connected in the given graph. We investigate the complexity of finding stable outcomes in such games, for several notions of stability. In particular, we provide an efficient algorithm that finds an individually stable partition for an arbitrary hedonic game on an acyclic graph. We also introduce a new stability concept -in-neighbor stability- which is tailored for our setting. We show that the problem of finding an in-neighbor stable outcome admits a polynomial-time algorithm if the underlying graph is a path, but is NP-hard for arbitrary trees even for additively separable hedonic games; for symmetric additively separable games we obtain a PLS-hardness result

Stable Roommate Problem with Diversity Preferences

In the multidimensional stable roommate problem, agents have to be allocated to rooms and have preferences over sets of potential roommates. We study the complexity of finding good allocations of agents to rooms under the assumption that agents have diversity preferences [Bredereck et al., 2019]: each agent belongs to one of the two types (e.g., juniors and seniors, artists and engineers), and agents' preferences over rooms depend solely on the fraction of agents of their own type among their potential roommates. We consider various solution concepts for this setting, such as core and exchange stability, Pareto optimality and envy-freeness. On the negative side, we prove that envy-free, core stable or (strongly) exchange stable outcomes may fail to exist and that the associated decision problems are NP-complete. On the positive side, we show that these problems are in FPT with respect to the room size, which is not the case for the general stable roommate problem. Moreover, for the classic setting with rooms of size two, we present a linear-time algorithm that computes an outcome that is core and exchange stable as well as Pareto optimal. Many of our results for the stable roommate problem extend to the stable marriage problem.Comment: accepted to IJCAI'2

Computational Complexity in Additive Hedonic Games

We investigate the computational complexity of several decision problems in hedonic coalition formation games and demonstrate that attaining stability in such games remains NP-hard even when they are additive. Precisely, we prove that when either core stability or strict core stability is under consideration, the existence problem of a stable coalition structure is NP-hard in the strong sense. Furthermore, the corresponding decision problems with respect to the existence of a Nash stable coalition structure and of an individually stable coalition structure turn out to be NP-complete in the strong sense

Optimal Partitions in Additively Separable Hedonic Games

We conduct a computational analysis of fair and optimal partitions in additively separable hedonic games. We show that, for strict preferences, a Pareto optimal partition can be found in polynomial time while verifying whether a given partition is Pareto optimal is coNP-complete, even when preferences are symmetric and strict. Moreover, computing a partition with maximum egalitarian or utilitarian social welfare or one which is both Pareto optimal and individually rational is NP-hard. We also prove that checking whether there exists a partition which is both Pareto optimal and envy-free is $\Sigma_{2}^{p}$-complete. Even though an envy-free partition and a Nash stable partition are both guaranteed to exist for symmetric preferences, checking whether there exists a partition which is both envy-free and Nash stable is NP-complete.Comment: 11 pages; A preliminary version of this work was invited for presentation in the session `Cooperative Games and Combinatorial Optimization' at the 24th European Conference on Operational Research (EURO 2010) in Lisbo