Sub-1.5 Time-Optimal Multi-Robot Path Planning on Grids in Polynomial Time
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Graph-based multi-robot path planning (MRPP) is NP-hard to optimally solve. In this work, we propose the first low polynomial-time algorithm for MRPP achieving 1–1.5 asymptotic optimality guarantees on solution makespan for random instances under very high robot density. Specifically, on an m_1× m_2 gird, m_1 ≥ m_2, our RTH (Rubik Table with Highways) algorithm computes solutions for routing up to m_1m_2/3 robots with uniformly randomly distributed start and goal configurations with a makespan of m_1 + 2m_2 + o(m_1), with high probability. Because the minimum makespan for such instances is m_1 + m_2 - o(m_1), also with high probability, RTH guarantees m_1+2m_2/m_1+m_2 optimality as m_1 →∞ for random instances with up to 1/3 robot density, with high probability. m_1+2m_2/m_1+m_2∈ (1, 1.5]. Alongside the above-mentioned key result, we also establish: (1) for completely filled grids, i.e., m_1m_2 robots, any MRPP instance may be solved in polynomial time under a makespan of 7m_1 + 14m_2, (2) for m_1m_2/3 robots, RTH solves arbitrary MRPP instances with makespan of 3m_1+4m_2 + o(m_1), (3) for m_1m_2/2 robots, a variation of RTH solves a random MRPP instance with the same 1-1.5 optimality guarantee, and (4) the same m_1+2m_2/m_1+m_2 optimality guarantee holds for regularly distributed obstacles at 1/9 density together with 2m_1m_2/9 randomly distributed robots; such settings directly map to real-world parcel sorting scenarios. In extensive numerical evaluations, RTH and its variants demonstrate exceptional scalability as compared with methods including ECBS and DDM, scaling to over 450 × 300 grids with 45,000 robots, and consistently achieves makespan around 1.5 optimal or better, as predicted by our theoretical analysis.
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