Sub-1.5 Time-Optimal Multi-Robot Path Planning on Grids in Polynomial Time

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 $m_1\times m_2$ gird, $m_1 \ge m_2$, our RTH (Rubik Table with Highways) algorithm computes solutions for routing up to $\frac{m_1m_2}{3}$ 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 $\frac{m_1+2m_2}{m_1+m_2}$ optimality as $m_1 \to \infty$ for random instances with up to $\frac{1}{3}$ robot density, with high probability. $\frac{m_1+2m_2}{m_1+m_2} \in (1, 1.5]$. 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 $\frac{m_1m_2}{3}$ robots, RTH solves arbitrary MRPP instances with makespan of $3m_1+4m_2 + o(m_1)$, (3) for $\frac{m_1m_2}{2}$ robots, a variation of RTH solves a random MRPP instance with the same 1-1.5 optimality guarantee, and (4) the same $\frac{m_1+2m_2}{m_1+m_2}$ optimality guarantee holds for regularly distributed obstacles at $\frac{1}{9}$ density together with $\frac{2m_1m_2}{9}$ 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 $450 \times 300$ grids with $45,000$ robots, and consistently achieves makespan around $1.5$ optimal or better, as predicted by our theoretical analysis.