Hedonic Diversity Games: A Complexity Picture with More than Two Colors

TL;DR

This work extends hedonic games to Hedonic Diversity Games with multiple colors, focusing on stable coalition structures under Nash and individual stability. It provides a complete parameterized complexity landscape, showing fixed-parameter tractability when the number of colors and the maximum coalition size are jointly bounded, and XP results for several other parameterizations, supported by ILP, DP, and network-flow techniques. The authors prove a suite of hardness results (NP-hard and W[1]-hard) via reductions from classic problems, establishing that color-bound is essential for tractability and resolving an open question for the two-color case. The findings offer both algorithmic pathways and tight hardness barriers across a broad, parameterized view of HDG, with implications for modeling diversity and fairness in coalition formation. The work also introduces Own-HDG as a restricted palette variant and discusses its distinct complexity behavior and future research directions, including other stability notions and broader parameter considerations.

Abstract

Hedonic diversity games are a variant of the classical Hedonic games designed to better model a variety of questions concerning diversity and fairness. Previous works mainly targeted the case with two diversity classes (represented as colors in the model) and provided some initial complexity-theoretic and existential results concerning Nash and individually stable outcomes. Here, we design new algorithms accompanied with lower bounds which provide a complete parameterized-complexity picture for computing Nash and individually stable outcomes with respect to the most natural parameterizations of the problem. Crucially, our results hold for general Hedonic diversity games where the number of colors is not necessarily restricted to two, and show that -- apart from two trivial cases -- a necessary condition for tractability in this setting is that the number of colors is bounded by the parameter. Moreover, for the special case of two colors we resolve an open question asked in previous work (Boehmer and Elkind, AAAI 2020).

Figures (3)

• Figure 1: The complexity picture for HDG for both Nash and individual stability. Combinations of parameters which give rise to fixed-parameter algorithms are highlighted in green, while combinations for which HDG is W[1]-hard but in XP are highlighted in orange and NP-complete combinations are highlighted in red. Results explicitly proved in this work are represented by a black box and a reference to the given theorem, corollary or lemma.
• Figure 2: Illustration of the network flow instance used in the proof of \ref{['lem:allHDG:XP:WeakNumSize']}. Agents' membership in sets of a certain color set are indicated by rectangles; correspondingly the coalition color tuple's second entries are indicated by the rounded rectangles. All except the thick edges have unit capacity, and the capacity on the thick edges is determined by the branching. The existence of edges in the middle is determined by the branching as well. There will never be edges between agents of a certain color and coalition color tuple with a different second entry, e.g. there will never be an edge between $b$ and $X$ or $Y$. In this example, the coalition that is the first entry of $X$ is valid for $a$, while the one of $Y$ is not.
• Figure 3: Exemplary illustration of the constructed HDG instances in the proof of \ref{['lem:allHDG:Wh:ColorsNumTypes']} with $k = 2$. The textbubbles give the coalitions which the respective agents would prefer to be included in rather than being alone. Each marker (colored with one of $\omega$ marker colors, indicated as different shades of blue) is associated to one of $S_1, \dotsc, S_\omega$ corresponding to its preferred coalitions; in this example $S_1$ consists of three vectors $(4,3),(1,7),(2,3)$, and $S_\omega$ consists of only one vector $(3,5)$. The normal agents encode $t$; in this example $t = (7,12)$.

Theorems & Definitions (32)

• Example 1
• Lemma 1
• proof
• proof
• Theorem 3
• proof
• Claim 1
• Theorem 4
• proof
• Theorem 5