Settling the Complexity of Popularity in Additively Separable and Fractional Hedonic Games

TL;DR

This work proves that deciding about the existence of popular partitions in additively separable and fractional hedonic games is $\Sigma_2^p$-complete, which is the first work that proves completeness of popularity for the second level of the polynomial hierarchy.

Abstract

We study coalition formation in the framework of hedonic games. There, a set of agents needs to be partitioned into disjoint coalitions, where agents have a preference order over coalitions. A partition is called popular if it does not lose a majority vote among the agents against any other partition. Unfortunately, hedonic games need not admit popular partitions and prior work suggests significant computational hardness. We confirm this impression by proving that deciding about the existence of popular partitions in additively separable and fractional hedonic games is $Σ_2^p$-complete. This settles the complexity of these problems and is the first work that proves completeness of popularity for the second level of the polynomial hierarchy.

Figures (4)

• Figure 1: Complexity hierarchy of popularity in coalitional scenarios. Gray boxes refer to our main results.
• Figure 2: A No-instance of Ashg-Popularity-Existence. Omitted edges imply value $-\infty$.
• Figure 3: The reduction for the proof of \ref{['thm_ashg']}. Omitted edges imply value $-\infty$. When two values $v_1/v_2$ appear, $v_1$ refers to corresponding agents, and $v_2$ to noncorresponding. Left-side agents are marked in blue. $b_1$ and $b_2$ are single agents, while the rest represent sets of agents.
• Figure 4: The reduction for the proof of \ref{['thm_fhg']}. Each node refers to a certain agent type, i.e., to the respective set of agents. Edges indicate valuations between all agents in the respective sets. When two values $v_1/v_2$ appear, $v_1$ refers to corresponding agents, and $v_2$ to noncorresponding ones. Omitted edges imply value $0$.

Theorems & Definitions (114)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1: BrBu20a, Proposition 4
• Lemma A.1
• proof
• Lemma A.2
• proof
• Lemma A.3
• proof
• Lemma A.4