This paper studies the allocation of indivisible items to agents, when each agent’s preferences are expressed by means of a directed acyclic graph. The vertices of each preference graph represent the subset of items approved of by the respective agent. An arc (a,b) in such a graph means that the respective agent prefers item a over item b. We introduce a new measure of dissatisfaction of an agent by counting the number of non-assigned items which are approved of by the agent and for which no more preferred item is allocated to the agent. Considering two problem variants, we seek an allocation of the items to the agents in a way that minimizes (i) the total dissatisfaction over all agents or (ii) the maximum dissatisfaction among the agents. For both optimization problems we study the status of computational complexity and obtain NP-hardness results as well as polynomial algorithms with respect to natural underlying graph structures, such as stars, trees, paths, and matchings. We also analyze the parameterized complexity of the two problems with respect to various parameters related to the number of agents, the dissatisfaction threshold, the vertex degrees of the preference graphs, and the treewidth.

Abstract In a hedonic diversity game (HDG) there are two types of agents (red and blue agents) that need to form disjoint coalitions, i.e., subgroups of agents. Each agent’s preferences over the coalitions depend on the relative number of agents of the same type in her coalition. In the special case of a dichotomous hedonic diversity game (DHDG) each agent distinguishes between approved and disapproved fractions only. We aim at outcomes that are stable against agents’ deviations, and at outcomes that maximize social welfare. In particular, we show that the strict core of a DHDG may be empty even in instances with only three agents, while each HDG with two agents has a non-empty strict core. We also provide several computational complexity results for DHDGs with respect to the number of fractions approved per agent. For instance, we prove that deciding whether a DHDG has a non-empty strict core is $$\textsf {NP}$$ NP -complete even when each agent approves of at most three fractions. In addition, we show that deciding whether a DHDG admits a Nash stable outcome is $$\textsf {NP}$$ NP -complete even in restricted settings with only two approved fractions per agent—therewith, improving a result in the literature. For the task of maximizing social welfare, we apply approval scores and Borda scores from voting theory. For DHDGs and approval scores, we draw the sharp separation line between polynomially solvable and $$\textsf {NP}$$ NP -complete cases with respect to the fixed number of approved fractions per agent. We complement these findings with an $$\textsf {NP}$$ NP -completeness result for HDGs under Borda scores.