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Team Coordination on Graphs: Problem, Analysis, and Algorithms

Yanlin Zhou, Manshi Limbu, Gregory J. Stein, Xuan Wang, Daigo Shishika, Xuesu Xiao

TL;DR

This work formalizes and analyzes Team Coordination on Graphs with Risky Edges (tcgre), showing the problem is NP-hard via a reduction from Maximum 3D Matching. It reexpresses tcgre as a constrained optimization and introduces a decomposition framework that yields three algorithm classes: (i) joint-state graph (jsg)-based methods with optimality guarantees, (ii) coordination-exhaustive search (ces) for near-optimal coordination with tractable complexity, and (iii) receding-horizon sub-team methods (rhoc-a*) that balance efficiency and solution quality. The authors provide both theoretical insights and extensive experiments demonstrating the tradeoffs between optimality and scalability, with rhoc-a* offering favorable performance for larger instances and ces providing strong guarantees under practical assumptions. The findings offer practical guidance for scalable, risk-aware multi-robot coordination and contribute a formal, decomposed approach to a challenging class of MAPF-like problems.

Abstract

Team Coordination on Graphs with Risky Edges (TCGRE) is a recently emerged problem, in which a robot team collectively reduces graph traversal cost through support from one robot to another when the latter traverses a risky edge. Resembling the traditional Multi-Agent Path Finding (MAPF) problem, both classical and learning-based methods have been proposed to solve TCGRE, however, they lacked either computational efficiency or optimality assurance. In this paper, we reformulate TCGRE as a constrained optimization problem and perform a rigorous mathematical analysis. Our theoretical analysis shows the NP-hardness of TCGRE by reduction from the Maximum 3D Matching problem and that efficient decomposition is a key to tackle this combinatorial optimization problem. Furthermore, we design three classes of algorithms to solve TCGRE, i.e., Joint State Graph (JSG) based, coordination based, and receding-horizon sub-team based solutions. Each of these proposed algorithms enjoy different provable optimality and efficiency characteristics that are demonstrated in our extensive experiments.

Team Coordination on Graphs: Problem, Analysis, and Algorithms

TL;DR

This work formalizes and analyzes Team Coordination on Graphs with Risky Edges (tcgre), showing the problem is NP-hard via a reduction from Maximum 3D Matching. It reexpresses tcgre as a constrained optimization and introduces a decomposition framework that yields three algorithm classes: (i) joint-state graph (jsg)-based methods with optimality guarantees, (ii) coordination-exhaustive search (ces) for near-optimal coordination with tractable complexity, and (iii) receding-horizon sub-team methods (rhoc-a*) that balance efficiency and solution quality. The authors provide both theoretical insights and extensive experiments demonstrating the tradeoffs between optimality and scalability, with rhoc-a* offering favorable performance for larger instances and ces providing strong guarantees under practical assumptions. The findings offer practical guidance for scalable, risk-aware multi-robot coordination and contribute a formal, decomposed approach to a challenging class of MAPF-like problems.

Abstract

Team Coordination on Graphs with Risky Edges (TCGRE) is a recently emerged problem, in which a robot team collectively reduces graph traversal cost through support from one robot to another when the latter traverses a risky edge. Resembling the traditional Multi-Agent Path Finding (MAPF) problem, both classical and learning-based methods have been proposed to solve TCGRE, however, they lacked either computational efficiency or optimality assurance. In this paper, we reformulate TCGRE as a constrained optimization problem and perform a rigorous mathematical analysis. Our theoretical analysis shows the NP-hardness of TCGRE by reduction from the Maximum 3D Matching problem and that efficient decomposition is a key to tackle this combinatorial optimization problem. Furthermore, we design three classes of algorithms to solve TCGRE, i.e., Joint State Graph (JSG) based, coordination based, and receding-horizon sub-team based solutions. Each of these proposed algorithms enjoy different provable optimality and efficiency characteristics that are demonstrated in our extensive experiments.
Paper Structure (23 sections, 1 theorem, 7 equations, 5 figures, 3 algorithms)

This paper contains 23 sections, 1 theorem, 7 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. mapf and Classes of Algorithms
  4. Team Coordination on Graphs with Risky Edges (tcgre)
  5. Problem Formulation
  6. Action & Cost Model
  7. Problem Definition
  8. Mathematical Analysis
  9. NP-Hardness
  10. Problem Analysis
  11. Solution Algorithms
  12. jsg-Based Solutions
  13. jsg Construction
  14. Sub-Problem 1
  15. Sub-Problem 2
  16. ...and 8 more sections

Key Result

Theorem 1

Maximum 3D Matching reduces to tcgre.

Figures (5)

  • Figure 1: Reduction from Maximum 3D Matching (Middle) to tcgre (Left and Right) and Inspiration for ces.
  • Figure 2: Receding-Horizon Optimistic Cooperative A*.
  • Figure 3: True and Naive Optimality vs. Time with jsg-ucs, jsg-a*, ces, rhoc-a*, and Naive. Each data point denotes the result for the experiment with # of Robots--# of Nodes. For visibility, cluttered areas are magnified in the dashed boxes.
  • Figure 4: ces Planning Time on Graphs with 2 Support Pairs.
  • Figure 5: rhoc-a* Planning Time.

Theorems & Definitions (2)

  • Definition 1
  • Theorem 1