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

TL;DR

This paper studies the computational complexity of forming teams of constrained maximum size in coalition formation problems, providing efficient algorithms for certain structures and proving their optimality.

Contribution

It introduces an FPT algorithm for bounded-size coalition formation on tree-like structures and proves its asymptotic optimality under theoretical assumptions.

Findings

Efficient FPT algorithm for treewidth-bounded structures

Proven asymptotic optimality of the algorithm

Intractability results for general cases

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}).