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

Abstract

It is well-known that 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 (i.e., the time it takes to complete a reconfiguration of the robots) for random instances under very high robot density, with high probability. Specifically, on an m1 ×m2 gird, m1 ≥ m2, our RTH (Rubik Table with Highways) algorithm computes solutions for routing up to m 1 m 2 3 robots with uniformly randomly distributed start and goal configurations with a makespan of m1 + 2m2 + o(m1), with high probability. Because the minimum makespan for such instances is m1 + m2 − o(m1), also with high probability, RTH guarantees m 1 +2m 2 m 1 +m 2 optimality as m1 → ∞ 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., m1m2 robots, any MRPP instance may be solved in polynomial time under a makespan of 7m1 + 14m2, (2) for m 1 m 2 3 robots, RTH solves arbitrary MRPP instances with makespan of 3m1 + 4m2 + o(m1), (3) for m 1 m 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 1 m 2 9 randomly distributed robots; such settings directly map to real-world parcel sorting scenarios.Moreover, we have developed effective, principled heuristics that further improve the computed optimality of RTH algorithms. 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.• The same m1+2m2 m1+m2 optimality guarantee may be achieved on grids with up to m1m2 9 regularly distributed obstacles together with 2m1m2 9 robots using RTH (e.g., Fig. 1(b)).

Polynomial Time Near-Time-Optimal Multi-Robot Path Planning in Three Dimensions with Applications to Large-Scale UAV Coordination

“…3). Such configurations are called balanced configurations [47]. In practice, configurations are likely not "far" from being balanced.…”

“…The matching heuristic we developed is based on linear bottleneck assignment (LBA), which runs in polynomial time. A 2D version of the heuristic is introduced in [47], to which readers are referred to for more details. Here, we illustrate how to apply LBA-matching in MatchingXY.…”

“…This research builds on [47], which addresses the MRPP problem in 2D. In comparison to [47], the current work describes a significant extension to the 3D (as well as higher dimensional) domain with many unique applications, including, e.g., the reconfiguration of large UAV fleets, the optimal coordination of air traffic, and the adjustment of satellite constellations.…”

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

“…VII. Given the amount of material included in the work, to provide a concentrated discussion, we refer the readers to [15] for proofs to theorems.…”