Exact Algorithms and Lowerbounds for Multiagent Pathfinding: Power of Treelike Topology

TL;DR

This paper studies the computational complexity of Multiagent Path Finding (MAPF), revealing its hardness and fixed-parameter tractability depending on graph structure and parameters like number of agents, graph diameter, and treewidth.

Contribution

It provides a comprehensive parameterized complexity analysis of MAPF, identifying conditions for fixed-parameter tractability and hardness based on graph parameters.

Findings

MAPF is W[1]-hard with respect to the number of agents and maximum degree.

MAPF remains NP-hard in planar graphs with fixed maximum degree and makespan.

An FPT algorithm is developed for parameters k and graph diameter.

Abstract

In the Multiagent Path Finding problem (MAPF for short), we focus on efficiently finding non-colliding paths for a set of $k$ agents on a given graph $G$, where each agent seeks a path from its source vertex to a target. An important measure of the quality of the solution is the length of the proposed schedule $\ell$, that is, the length of a longest path (including the waiting time). In this work, we propose a systematic study under the parameterized complexity framework. The hardness results we provide align with many heuristics used for this problem, whose running time could potentially be improved based on our fixed-parameter tractability results. We show that MAPF is W[1]-hard with respect to $k$ (even if $k$ is combined with the maximum degree of the input graph). The problem remains NP-hard in planar graphs even if the maximum degree and the makespan$\ell$ are fixed constants. On the positive side, we show an FPT algorithm for $k+\ell$. As we delve further, the structure of~$G$ comes into play. We give an FPT algorithm for parameter $k$ plus the diameter of the graph~$G$. The MAPF problem is W[1]-hard for cliquewidth of $G$ plus $\ell$ while it is FPT for treewidth of $G$ plus $\ell$.