An Optimization Variant of Multi-Robot Path Planning Is Intractable

Abstract

An optimization variant of a problem of path planning for multiple robots is addressed in this work. The task is to find spatial-temporal path for each robot of a group of robots such that each robot can reach its destination by navigating through these paths. In the optimization variant of the problem, there is an additional requirement that the makespan of the solution must be as small as possible. A proof of the claim that optimal path planning for multiple robots is NP‑complete is sketched in this short paper.

Optimal and Bounded-Suboptimal Multi-Goal Task Assignment and Path Finding

“…Many problems that are related to our problem have been proposed and studied in recent years. MAPF: The MAPF problem is NP-hard to solve optimally for flowtime (the sum of the finish times of all agents in the last goal locations of their assigned tasks) minimization [8] and even NP-hard to approximate within any constant factor less than 4/3 for makespan (the maximum of the finish times of all agents in their pre-assigned goal locations) minimization [9], [10]. MAPF algorithms include reductions to other well-studied optimization problems [11]- [13] and specialized rule-based, search-based and hybrid algorithms [14]- [20].…”

Solution Quality Improvements for Massively Multi-Agent Pathfinding

“…We provide a high-level description of the previous MAPP algorithm, focusing on features relevant to our work. More details can be found in Wang and Botea (2009;2010).…”

“…In multi-agent pathfinding, MAPP (Wang and Botea 2009;2010) has previously been shown to be state-of-the-art in terms of scalability and success ratio (i.e., percentage of solved units), on problems involving significantly larger numbers of mobile units than can be tractably handled using optimal algorithms. MAPP further provides a formal characterization of problems it can solve, and low-polynomial upper bounds on the resources required.…”

“…Since finding optimal solutions is NP-complete (Surynek 2010;Ratner and Warmuth 1986), optimal algorithms have limited scalability. To evaluate the quality of the solutions provided by suboptimal algorithms, we compare their solutions to lower bounds of optimal values, which are cheap to compute.…”

Improving Continuous-time Conflict Based Search

“…Many real-world applications require solving variants of MAPF, including managing aircraft-towing vehicles (Morris et al 2016), video game characters (Silver 2005), office robots (Veloso et al 2015), and warehouse robots (Wurman, D'Andrea, and Mountz 2007). Solving MAPF optimally (for common objective functions) is NP-Hard (Surynek 2010;Yu and LaValle 2013), but modern optimal MAPF algorithms can scale to problems with over a hundred agents (Sharon et al 2015;Boyarski et al 2015;Felner et al 2018;Lam et al 2019;Gange, Harabor, and Stuckey 2019;Surynek et al 2016).…”