Refined Hardness of Distance-Optimal Multi-Agent Path Finding

Abstract

**Read paper on the following link:** https://ifaamas.org/Proceedings/aamas2022/pdfs/p481.pdf **Abstract:** We study the computational complexity of multi-agent path finding (MAPF) on grid graphs. Given a graph $G$ and a set of agents, each having a start and target vertex, the goal is to find collision-free paths minimizing the total distance traveled. Formulations restricting $G$ to be a 2D grid are ubiquitous, as it conveniently allows for modeling well-structured environments (e.g., warehouses). Previous hardness results had $G$ either (1) be a 2D grid or (2) have more than one vertex unoccupied by a agent, but never considered both properties simultaneously. The most restricted result for property (2) was shown for (non-grid) planar graphs. We refine previous results by proving that distance-optimal MAPF remains NP-hard for both properties simultaneously, thereby settling an open problem posed in Banfi et al. (2017). We present a reduction directly from 3SAT using simple gadgets, making our proof arguably more comprehensible and informative. We explain the potential benefits of these qualities towards positive results. Furthermore, our result is the first linear-time/size reduction for the case where $G$ is planar, appearing nearly four decades after the first related result. This allows us to go a step further and exploit the Exponential Time Hypothesis (ETH) to obtain an exponential lower bound for the running time. To the best of our knowledge, this is the first application of ETH to show \textit{any} concrete lower bound for MAPF. Finally, as a stepping stone, we also show hardness for the monotone case, in which agents move one by one with no intermediate stops.