A New Upper Bound for the Makespan of Cost-Optimal Solutions for Multi-Agent Path Finding (Extended Abstract)

Abstract

A well-known approach to solving Multi-Agent Path Finding (MAPF) optimally is compilation to Boolean Satisfiability or Answer Set Programming (ASP). Such compilation-based approaches are superior to other approaches on dense, relatively small instances and may invoke the solver multiple times, each with an encoding of the same instance for a different makespan. Critical to their performance is the runtime of the last solver invocation, whose input is the instance encoded with a theoretical upper bound of the makespan of the optimal solution. In this paper, we propose a new theoretical upper bound for such a last invocation. Unlike the previously known bound, when given a MAPF instance P, our bound requires a solution to P_1, a version of P where one of its agents is removed. We prove that our bound is correct and experimentally significantly tighter than the previously known bound. We propose a recursive parallel approach that allows us to exploit our new bound effectively. Our evaluation of warehouses and random MAPF benchmarks of varied sizes shows that our bound is, on average, 21.2% smaller than the previous bound. This allows for generating grounded ASP formulas around 33.45% smaller and solving 4.9% more instances.