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

2025-04-11

AAAI Conference on Artificial Intelligence

AI Summary

This paper studies coalition formation with an upper bound on coalition size: partitioning agents into efficient coalitions while respecting their preferences over potential teammates. As this problem is NP-hard, we present the first progressively optimal fixed-parameter tractable (FPT) algorithm, parameterized by treewidth and vertex cover number. We rigorously establish tightness of its time complexity under both parameters—proving that no significantly faster FPT algorithm exists unless standard complexity assumptions fail. Experimental evaluation demonstrates strong scalability and efficiency on tree-structured instances. Our core contribution is the first progressively optimal FPT framework for bounded-size coalition formation, complemented by combinatorial lower-bound techniques that formally certify its theoretical optimality boundary.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Imagine we want to split a group of agents into teams in the most efficient way, considering that each agent has their own preferences about their teammates. This scenario is modeled by the extensively studied 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 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 vertex cover number).

Problem

• Study coalition formation with bounded team sizes
• Provide exact algorithms and intractability results
• Optimize tree-like structures for small teams

Innovation

• Exact algorithms for bounded team sizes
• Efficient tree-like structure handling
• Asymptotically optimal algorithm proof