Coalition Formation with Bounded Coalition Size

TL;DR

This work studies Additively Separable Hedonic Games with bounded coalition size, focusing on MaxUtil optimization and stability concepts under a size cap $k$. It introduces the MnM algorithm, a polynomial-time approximation that achieves $1/(k-1)$-approximation in the unweighted case and parity-based guarantees in the weighted case, with runtime $O(k n^{2.5})$. The paper also analyzes the existence and complexity of stable partitions: the Contractual Strict Core is always nonempty and computable in polynomial time, while the Strict Core may be empty and its existence NP-hard to decide; for the unweighted case, the Core is nonempty at $k=3$ with a polynomial core-finding algorithm, but the weighted Core can be empty and is NP-hard to determine. These results provide a comprehensive treatment of efficiency and stability for bounded coalitions in symmetric ASHGs and identify several open problems and directions for future work.

Abstract

In many situations when people are assigned to coalitions, the utility of each person depends on the friends in her coalition. Additionally, in many situations, the size of each coalition should be bounded. This paper studies such coalition formation scenarios in both weighted and unweighted settings. Since finding a partition that maximizes the utilitarian social welfare is computationally hard, we provide a polynomial-time approximation algorithm. We also investigate the existence and the complexity of finding stable partitions. Namely, we show that the Contractual Strict Core (CSC) is never empty, but the Strict Core (SC) of some games is empty. Finding partitions that are in the CSC is computationally easy, but even deciding whether an SC of a given game exists is NP-hard. The analysis of the core is more involved. In the unweighted setting, we show that when the coalition size is bounded by 3 the core is never empty, and we present a polynomial time algorithm for finding a member of the core. However, for the weighted setting, the core may be empty, and we prove that deciding whether there exists a core is NP-hard.

Figures (8)

• Figure 1: An example for the MaxUtil problem where $k=3$.
• Figure 2: An example for Algorithm \ref{['alg:MnM']} where $k=4$.
• Figure 3: Examples for edge coloring in graphs with an odd and an even number of vertices.
• Figure 4: An example of a weighted graph in which the core is empty, for $k=3$.
• Figure 5: An illustration of the construction in Theorem \ref{['thm:core_np_w']} for $k=5$, showing the edges between $T$, $D_s$ for some $s \in \{1, \dots, n + 1\}$ and $u_i$ for some $i \in A$. Additionally, we show the edge between $u_i$ and an arbitrary $u_j, j \in A$.

Theorems & Definitions (20)

• Definition 4.1: MaxUtil problem
• Lemma 1
• Lemma 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 5.1: Core existence problem
• Definition 5.2: 3-Dimensional Stable Roommates with Metric Preferences (Metric-3DSR)
• Theorem 6
• Definition 5.3: $SC$ existence problem