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Robust Multiagent Collaboration Through Weighted Max-Min T-Joins

Sharareh Alipour

TL;DR

This work tackles robust multiagent group formation by studying the weighted max-min T-join problem, formalized as maximizing the minimum-cost matching over all even-sized vertex subsets. It introduces three main contributions: a farthest-point greedy upper bound yielding a $2\ln n$-approximation for the general problem with $O(n^4)$ running time, an ear-decomposition based upper bound for weighted max-min $T$-joins, and an exact algorithm for $(1,2)$-weighted graphs, complemented by empirical evidence of tight bounds on real datasets. The results demonstrate both theoretical significance and practical utility for fair, robust pairing and coalition formation in multiagent systems, with strong performance observed when combining lower and upper bounds. Overall, the methods provide implementable, scalable tools for resilient task assignment and resource sharing in networked agent populations.

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.

Robust Multiagent Collaboration Through Weighted Max-Min T-Joins

TL;DR

This work tackles robust multiagent group formation by studying the weighted max-min T-join problem, formalized as maximizing the minimum-cost matching over all even-sized vertex subsets. It introduces three main contributions: a farthest-point greedy upper bound yielding a -approximation for the general problem with running time, an ear-decomposition based upper bound for weighted max-min -joins, and an exact algorithm for -weighted graphs, complemented by empirical evidence of tight bounds on real datasets. The results demonstrate both theoretical significance and practical utility for fair, robust pairing and coalition formation in multiagent systems, with strong performance observed when combining lower and upper bounds. Overall, the methods provide implementable, scalable tools for resilient task assignment and resource sharing in networked agent populations.

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 -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 -matching problem}, where the chosen set must contain exactly vertices. Building on this bound, we develop a general algorithm with a \emph{-approximation guarantee} that runs in time. Second, using ear decompositions, we propose another upper bound for the weighted max-min -join cost. We also show that the problem can be solved exactly when edge weights belong to . 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 -joins as a framework for fair and robust group formation in multiagent systems.
Paper Structure (16 sections, 6 theorems, 18 equations, 1 figure, 4 tables)

This paper contains 16 sections, 6 theorems, 18 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Our approach and new results
  4. Notations.
  5. From the weighted max--min $2k$-matching problem to the weighted max-min $T$-join
  6. Ear decomposition and weighted max–min $T$-join problem
  7. Remark.
  8. Exact algorithm for weighted max–min $T$-join in $(1,2)$-graphs
  9. Remark.
  10. Experimental Evaluation
  11. Reviewer assignment
  12. Graph construction.
  13. Lower bound (Theorem \ref{['main11']}).
  14. Upper bound via ear decomposition (Theorem \ref{['upperbound']}).
  15. Fair Mutual-Aid Pairing for Mobile Services
  16. ...and 1 more sections

Key Result

theorem 1

Suppose $\{v'_1, \dots, v'_{2k}\} \subseteq V(G)$ is an optimal set for the weighted max--min $2k$-matching problem, i.e., $\mu_{2k}(G) = mwm(v'_1, \dots, v'_{2k})$ Then,

Figures (1)

  • Figure 1: An example of a graph $G$ such that $\mu(G)=9+\epsilon\neq \sum \max(P_i^*)=9+2\epsilon$, where $\epsilon$ is a small constant number .

Theorems & Definitions (6)

  • theorem 1
  • theorem 2
  • theorem 3: lovasz1972note
  • lemma 1
  • theorem 4
  • theorem 5