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. The dual guarantee on computational efficiency and solution optimality suggests our proposed general method is promising in significantly scaling up multi-robot applications for logistics, e.g., at large robotic warehouses.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.Alongside this key result, we also establish a series of related results supporting even higher robot densities and environments with regularly distributed obstacles, which directly map to real-world parcel sorting scenarios. Building on the baseline methods with provable guarantees, we have developed effective, principled heuristics that further improve the computed optimality of the 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.
In this work, we systematically examine the application of spatio-temporal splitting heuristics to the Multi-Robot Motion Planning (MRMP) problem in a graph-theoretic setting: a problem known to be NP-hard to optimally solve. Following the divide-and-conquer principle, we design multiple spatial and temporal splitting schemes that can be applied to any existing MRMP algorithm, including integer programming solvers and Enhanced Conflict Based Search, in an orthogonal manner. The combination of a good baseline MRMP algorithm with a proper splitting heuristic proves highly effective, allowing the resolution of problems 10+ times than what is possible previously, as corroborated by extensive numerical evaluations. Notably, spatial partition of problem fusing with the temporal splitting heuristic and the enhanced conflict based search (ECBS) algorithm increases the scalability of ECBS on large and challenging DAO maps by 5-15 folds with negligible impact on solution optimality.
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