Multi-agent pathfinding (MAPF) traditionally focuses on collision avoidance, but many real-world applications require active coordination between agents to improve team performance. This paper introduces Team Coordination on Graphs with Risky Edges (TCGRE), where agents collaborate to reduce traversal costs on high-risk edges via support from teammates. We reformulate TCGRE as a 3D matching problem-mapping robot pairs, support pairs, and time steps-and rigorously prove its NP-hardness via reduction from Minimum 3D Matching. To address this complexity, (in the conference version) we proposed efficient decomposition methods, reducing the problem to tractable subproblems: Joint-State Graph (JSG): Encodes coordination as a single-agent shortest-path problem. Coordination-Exhaustive Search (CES): Optimizes support assignments via exhaustive pairing. Receding-Horizon Optimistic Cooperative A* (RHOCA*): Balances optimality and scalability via horizon-limited planning. Further in this extension, we introduce a dynamic graph construction method (Dynamic-HJSG), leveraging agent homogeneity to prune redundant states and reduce computational overhead by constructing the joint-state graph dynamically. Theoretical analysis shows Dynamic-HJSG preserves optimality while lowering complexity from exponential to polynomial in key cases. Empirical results validate scalability for large teams and graphs, with HJSG outperforming baselines greatly in runtime in different sizes and types of graphs. This work bridges combinatorial optimization and multi-agent planning, offering a principled framework for collaborative pathfinding with provable guarantees, and the key idea of the solution can be widely extended to many other collaborative optimization problems, such as MAPF.

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 computation efficiency or optimality assurance. In this paper, we reformulate TCGRE as a constrained optimization and perform 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. Further more, 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.