Hedonic games provide a model of coalition formation in which a set of agents is partitioned into coalitions and the agents have preferences over which set they belong to. Recently, Aziz et. al. (2014) have initiated the study of hedonic games with dichotomous preferences, where each agent either approves or disapproves of a given coalition. In this work, we study the computational complexity of questions related to finding optimal and stable partitions in dichotomous hedonic games under various ways of restricting and representing the collection of approved coalitions. Encouragingly, many of these problems turn out to be polynomial-time solvable. In particular, we show that an individually stable outcome always exists and can be found in polynomial time. We also provide efficient algorithms for cases in which agents approve only few coalitions, in which they only approve intervals, and in which they only approve sets of size 2 (the roommates case). These algorithms are complemented by NP-hardness results, especially for representations that are very expressive, such as in the case when agents' goals are given by propositional formulas.

Preference profiles that are single-peaked on trees enjoy desirable properties: they admit a Condorcet winner (Demange 1982), and there are hard voting problems that become tractable on this domain (Yu et al., 2013). Trick (1989) proposed a polynomial-time algorithm that finds some tree with respect to which a given preference profile is single-peaked. However, some voting problems are only known to be easy for profiles that are single-peaked on "nice" trees, and Trick's algorithm provides no guarantees on the properties of the tree that it outputs. To overcome this issue, we build on the work of Trick and Yu et al. to develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is single-peaked. We show how to use this representation to efficiently find the "best" tree for a given profile, according to a number of criteria; for other criteria, we obtain NP-hardness results. In particular, we show that it is NP-hard to decide whether an input profile is single-peaked with respect to a given tree. To demonstrate the applicability of our framework, we use it to identify a new class of profiles that admit an efficient algorithm for a popular variant of the Chamberlin-Courant rule.

Hedonic games are a well-studied model of coalition formation, in which selfish agents are partitioned into disjoint sets and agents care about the make-up of the coalition they end up in. The computational problems of finding stable, optimal, or fair outcomes tend to be computationally intractable in even severely restricted instances of hedonic games. We introduce the notion of a graphical hedonic game and show that, in contrast, on classes of graphical hedonic games whose underlying graphs are of bounded treewidth and degree, such problems become easy. In particular, problems that can be specified through quantification over agents, coalitions, and (connected) partitions can be decided in linear time. The proof is by reduction to monadic second order logic. We also provide faster algorithms in special cases, and show that the extra condition of the degree bound cannot be dropped. Finally, we note that the problem of allocating indivisible goods can be modelled as a hedonic game, so that our results imply tractability of finding fair and efficient allocations on appropriately restricted instances.

Hedonic games are a well-studied model of coalition formation, in which selfish agents are partitioned into disjoint sets, and agents care about the make-up of the coalition they end up in. The computational problem of finding a stable outcome tends to be computationally intractable, even after severely restricting the types of preferences that agents are allowed to report. We investigate a structural way of achieving tractability, by requiring that agents' preferences interact in a well-behaved manner. Precisely, we show that stable outcomes can be found in linear time for hedonic games that satisfy a notion of bounded treewidth and bounded degree.

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.