Optimal Multi-agent Path Finding in Continuous Time

Abstract

Continuous-time Conflict Based-Search (CCBS) has long been viewed as the standard optimal baseline for multi-agent path finding in continuous time (MAPFR), yet recent critiques show that the theoretically described CCBS can fail to terminate on solvable MAPFR problems while the publicly available reference implementation can return sub-optimal solutions. This work presents an analytical framework that yields simple and sufficient conditions under which any CCBS-style algorithm is both sound and solution complete. Investigating the reference CCBS implementation reveals that it violates our sufficient conditions for soundness, with counterexamples demonstrating sub-optimality. Leveraging the framework, we introduce a branching rule ($δ$-BR) and prove it restores soundness and termination guarantees. Consequently, the resulting CCBS variant is both sound and solution complete. To our knowledge, this is the first MAPFR solver matching the guarantees of the discrete-time CBS. On a constructed example, CCBS with $δ$-BR improves sum-of-costs from 10.707 to 9.000 ($\approx$ 16% lower) compared to the reference CCBS implementation. Across benchmarks, the reference CCBS implementation is generally able to find solutions faster than CCBS with $δ$-BR due to its more aggressive pruning. However, this comes at the cost of occasional sub-optimality and potential non-termination when all solutions are pruned, whereas $δ$-BR preserves optimality and guarantees termination by design. Because $δ$-BR largely only affects the branching step, it can be adopted as a drop-in replacement in existing codebases. Beyond CCBS, the analytical framework and termination criterion provide a systematic way to evaluate other CCBS-like MAPFR solvers and future extensions, thereby offering tools for rigorous analysis of next-generation MAPFR algorithms.