Structure and Intractability of Optimal Multi-Robot Path Planning on Graphs

Abstract

In this paper, we study the structure and computational complexity of optimal multi-robot path planning problems on graphs. Our results encompass three formulations of the discrete multi-robot path planning problem, including a variant that allows synchronous rotations of robots along fully occupied, disjoint cycles on the graph. Allowing rotation of robots provides a more natural model for multi-robot path planning because robots can communicate.Our optimality objectives are to minimize the total arrival time, the makespan (last arrival time), and the total distance. On the structure side, we show that, in general, these objectives demonstrate a pairwise Pareto optimal structure and cannot be simultaneously optimized. On the computational complexity side, we extend previous work and show that, regardless of the underlying multi-robot path planning problem, these objectives are all intractable to compute. In particular, our NP-hardness proof for the time optimal versions, based on a minimal and direct reduction from the 3-satisfiability problem, shows that these problems remain NP-hard even when there are only two groups of robots (i.e. robots within each group are interchangeable).

Multiagent Connected Path Planning: PSPACE-Completeness and How to Deal With It

“…Consider now the following variant of MCPP able to model collisions among the agents: two or more agents are neither allowed to occupy the same vertex at the same time, nor to move along the same edge between two subsequent steps. This model generalizes the graph-based Multirobot Path Planning model of (Yu and LaValle 2013b). It is immediate to notice that our proof holds even for this case, since the agents of our reduction are forced to move on non-overlapping portions of G E .…”

“…Our theoretical result implies that, not only a polynomial-time feasibility algorithm is unlikely to exist (unless P=PSPACE), but also feasibility certificates of polynomial size might be out of reach (unless NP=PSPACE). We also consider a variation of MCPP able to model collisions among agents, generalizing the graphbased Multirobot Path Planning problem (MPP) introduced by (Yu and LaValle 2013b), and show that our proof holds even in this case. This is interesting, since the MPP feasibility decision problem is in P (Yu and Rus 2015), while MPP time-and distance-optimal decision problems are NPcomplete (Yu 2016;Banfi, Basilico, and Amigoni 2017).…”

Multi-Agent Path Finding with Payload Transfers and the Package-Exchange Robot-Routing Problem

“…Corollary 8 improves the state-of-the-art NP-hardness result of MAPF for makespan minimization (Yu and LaValle 2013b), which is based on reducing 2/2/4-SAT to the (n 2 − 1)-puzzle (Ratner and Warmuth 1990), since it shows not only the NP-hardness of solving MAPF but also the NPhardness of approximating it with constant-factor approximations. Their proof does not transfer to PERR.…”

Sub-1.5 Time-Optimal Multi-Robot Path Planning on Grids in Polynomial Time

“…The main objective of MRPP is to find a set of collision-free paths for routing many robots from a start configuration to a goal configuration. In practice, solution optimality is also of key importance; yet optimally solving MRPP is generally NP-hard [44,53], even in planar [50] and grid settings [10]. MRPP algorithms find many important largescale applications, including, e.g., in warehouse automation for general order fulfillment [49], grocery order fulfillment [33], and parcel sorting [48].…”

“…As such, in this paper, MRPP refers explicitly to graph-based multi-robot path planning. Whereas the feasibility question has long been positively answered for MRPP [24], the same cannot be said when it comes to securing optimal solutions, as computing time-or distance-optimal solutions are shown to be NP-hard in many settings, including for general graphs [14,44,53], planar graphs [50,2], and even regular grids [10], similar to the setting addressed in this study.…”