Robust Multiagent Collaboration Through Weighted Max-Min T-Joins

TL;DR

This paper introduces algorithms for the weighted max-min T-join problem, providing approximation guarantees and bounds, with practical evaluation demonstrating their effectiveness in multiagent collaboration scenarios.

Contribution

It develops new algorithms and bounds for the weighted max-min T-join problem, including a 2 ln n-approximation and exact solutions for specific weights, advancing robust multiagent group formation.

Findings

Algorithms achieve small constant-factor approximations in practice.

Upper and lower bounds are consistently close on real datasets.

The methods are effective for fair and robust multiagent collaboration.

Abstract

Many multiagent tasks -- such as reviewer assignment, coalition formation, or fair resource allocation -- require selecting a group of agents such that collaboration remains effective even in the worst case. The \emph{weighted max-min $T$-join problem} formalizes this challenge by seeking a subset of vertices whose minimum-weight matching is maximized, thereby ensuring robust outcomes against unfavorable pairings. We advance the study of this problem in several directions. First, we design an algorithm that computes an upper bound for the \emph{weighted max-min $2k$-matching problem}, where the chosen set must contain exactly $2k$ vertices. Building on this bound, we develop a general algorithm with a \emph{$2 \ln n$-approximation guarantee} that runs in $O(n^4)$ time. Second, using ear decompositions, we propose another upper bound for the weighted max-min $T$-join cost. We also show that the problem can be solved exactly when edge weights belong to $\{1,2\}$. Finally, we evaluate our methods on real collaboration datasets. Experiments show that the lower bounds from our approximation algorithm and the upper bounds from the ear decomposition method are consistently close, yielding empirically small constant-factor approximations. Overall, our results highlight both the theoretical significance and practical value of weighted max-min $T$-joins as a framework for fair and robust group formation in multiagent systems.