Optimal and Scalable MAPF via Multi-Marginal Optimal Transport and Schr\"odinger Bridges

TL;DR

This paper introduces a scalable, optimal multi-agent pathfinding method by reformulating it as a multi-marginal optimal transport problem and applying Schr"odinger bridges for large-scale solutions.

Contribution

It develops a novel framework connecting MAPF with MMOT and Schr"odinger bridges, enabling polynomial-time, near-optimal solutions for large-scale anonymous multi-agent problems.

Findings

MAPF can be formulated as a multi-marginal optimal transport problem.

The Schr"odinger bridge approach reduces complexity and yields near-optimal transports.

Experimental results demonstrate the method's scalability and optimality.

Abstract

We consider anonymous multi-agent path finding (MAPF) where a set of robots is tasked to travel to a set of targets on a finite, connected graph. We show that MAPF can be cast as a special class of multi-marginal optimal transport (MMOT) problems with an underlying Markovian structure, under which the exponentially large MMOT collapses to a linear program (LP) polynomial in size. Focusing on the anonymous setting, we establish conditions under which the corresponding LP is feasible, totally unimodular, and consequently, yields min-cost, integral $(\{0,1\})$ transports that do not overlap in both space and time. To adapt the approach to large-scale problems, we cast the MAPF-MMOT in a probabilistic framework via Schr\"odinger bridges. Under standard assumptions, we show that the Schr\"odinger bridge formulation reduces to an entropic regularization of the corresponding MMOT that admits an iterative Sinkhorn-type solution. The Schr\"odinger bridge, being a probabilistic framework, provides a shadow (fractional) transport that we use as a template to solve a reduced LP and demonstrate that it results in near-optimal, integral transports at a significant reduction in complexity. Extensive experiments highlight the optimality and scalability of the proposed approaches.