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.

翻译：许多多智能体任务——如审稿人分配、联盟形成或公平资源分配——需要选择一组智能体，使得即使在最坏情况下协作仍能保持有效。\emph{加权最大最小$T$-连接问题}通过寻找一个顶点子集，使其最小权重匹配最大化，从而形式化了这一挑战，确保即使在不利配对下也能获得鲁棒的结果。我们在多个方向上推进了该问题的研究。首先，我们设计了一种算法，用于计算\emph{加权最大最小$2k$-匹配问题}的上界，其中所选集合必须恰好包含$2k$个顶点。基于此上界，我们开发了一种通用算法，具有\emph{$2 \ln n$近似保证}，运行时间为$O(n^4)$。其次，利用耳分解，我们提出了加权最大最小$T$-连接成本的另一个上界。我们还证明了当边权重属于$\{1,2\}$时，该问题可以精确求解。最后，我们在真实协作数据集上评估了我们的方法。实验表明，来自我们近似算法的下界与来自耳分解方法的上界始终接近，从而在经验上实现了较小的常数因子近似。总体而言，我们的结果凸显了加权最大最小$T$-连接作为多智能体系统中公平且鲁棒的群体形成框架的理论意义和实用价值。