On SAT-Based Approaches for Multi-Agent Path Finding with the Sum-of-Costs Objective

Multi-Agent Path Finding (MAPF) deals with the problem of finding collision-free paths for a set of agents. Each agent moves from its start location to its destination location in a shared environment represented by a graph. Reductionbased solving approaches for MAPF, for example, reduction to SAT, exploit a time-expanded layered graph, where each layer corresponds to specific time. Hence, these approaches are natural for minimizing Makespan (the shortest time until all agents reach their destinations). Modeling the other frequently used objective, namely Sum of Costs (SoC; the sum of paths lengths of all agents) is more difficult as the solution with the smallest SoC may not be reached in the timeexpanded graph with the smallest Makespan. In this paper we suggest a novel approach to estimate the Makespan, that guarantees the existence of a SoC-optimal solution. We also propose a novel pre-processing technique reducing the number of variables in the SAT model. The approach is empirically compared with an existing reduction-based method as well as with the state-of-the-art search-based optimal MAPF solver.