Optimal and Scalable MAPF via Multi-Marginal Optimal Transport and Schrödinger Bridges

2026-05-11

AI Summary

This work addresses the challenges of computational complexity and optimality in large-scale anonymous multi-agent path finding (MAPF) by formulating MAPF for the first time as a multi-marginal optimal transport problem with Markovian structure. By proving the total unimodularity of the constraint matrix in the associated linear program, the approach guarantees conflict-free, exact integer solutions. To enable scalable computation, the authors introduce a Schrödinger bridge framework with entropy regularization, solved efficiently via Sinkhorn iterations. This method significantly reduces computational complexity while preserving spatiotemporal collision-free paths and maintaining either optimality or near-optimality. Empirical results demonstrate excellent scalability and solution efficiency in large-scale scenarios.

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 Schrödinger bridges. Under standard assumptions, we show that the Schrödinger bridge formulation reduces to an entropic regularization of the corresponding MMOT that admits an iterative Sinkhorn-type solution. The Schrödinger 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.

Problem

• multi-agent path finding
• anonymous MAPF
• multi-marginal optimal transport
• Schrödinger bridges
• scalability

Innovation

• Multi-Agent Path Finding
• Multi-Marginal Optimal Transport
• Schrödinger Bridges
• Entropic Regularization
• Total Unimodularity