Exact Algorithms and Lower Bounds for Forming Coalitions of Constrained Maximum Size

TL;DR

This work analyzes the C-Coalition Formation problem under bounded coalition size in Additive Separable Hedonic Games, focusing on parameterized complexity with respect to graph-structure parameters. It provides an FPT algorithm for Weighted $\mathcal{C}$-Coalition Formation parameterized by $tw+\mathcal{C}$ via a tree-decomposition DP, proves W[1]-hardness for tree-depth, and presents both kernelization results (polynomial kernel for unweighted with $\text{vc}+\mathcal{C}$ and a negative result for weighted) and ETH-based lower bounds. The authors further show FPT algorithms for vertex cover and vertex integrity parameters, and establish hardness results for twin-cover number, painting a detailed tractability map across structural graph parameters. The findings offer practical guidelines for coalition formation on tree-like or clique-limited networks and contribute to a nuanced understanding of the computational limits and opportunities in constrained coalition formation problems.

Abstract

Imagine we want to split a group of agents into teams in the most \emph{efficient} way, considering that each agent has their own preferences about their teammates. This scenario is modeled by the extensively studied \textsc{Coalition Formation} problem. Here, we study a version of this problem where each team must additionally be of bounded size. We conduct a systematic algorithmic study, providing several intractability results as well as multiple exact algorithms that scale well as the input grows (FPT), which could prove useful in practice. Our main contribution is an algorithm that deals efficiently with tree-like structures (bounded \emph{treewidth}) for ``small'' teams. We complement this result by proving that our algorithm is asymptotically optimal. Particularly, there can be no algorithm that vastly outperforms the one we present, under reasonable theoretical assumptions, even when considering star-like structures (bounded \emph{vertex cover number}).

Figures (6)

• Figure 1: Overview of our results. A parameter $A$ appearing linked to a parameter $B$ with $A$ being below $B$ is to be understood as "there is a function $f$ such that $f(A)\geq f(B)$". In blue (red resp.) we exhibit the FPT ($\textsf{W}[1]$-hardness resp.) results we provide. The clique number of the graph is denoted by $\omega$. Note that our FPT results are for the more general, weighted version of the problem ($\mathcal{C}$-CFw), while our $\textsf{W}[1]$-hardness results are for the more restricted, unweighted version of the problem ($\mathcal{C}$-CF). Finally, note that this figure does not include our results concerning the kernelization of $\mathcal{C}$-CF.
• Figure 2: An example of possible solutions to the unweighted version of the $\mathcal{C}$-Coalition Formation problem. The input consists of the graph $G$ illustrated in subfigure (a), and the capacity $\mathcal{C}=4$. The $4$-partition in subfigure (b) has the optimal value of 6. Observe that every possible $4$-partition that includes a set of $4$ vertices (some of which are not included here) is suboptimal.
• Figure 3: The gadgets used in the proof of Theorem \ref{['thm:treedepth']}
• Figure 4: The graph $G$ constructed in the proof of Theorem \ref{['thm:treedepth']}.
• Figure 5: The gadget $G_{i,j}$ used in the construction of Theorem \ref{['thm:vc-to-the-vc']}.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 3.1
• proof