In this paper, we study the structure and computational com-plexity of optimal multi-robot path planning problems on graphs. Our results encompass three formulations of the dis-crete multi-robot path planning problem, including a variant that allows synchronous rotations of robots along fully occu-pied, 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 ob-jectives demonstrate a pairwise Pareto optimal structure and cannot be simultaneously optimized. On the computational complexity side, we extend previous work and show that, re-gardless 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).

The task in the multi-agent path finding problem (MAPF) is to find paths for multiple agents, each with a different start and goal position, such that agents do not collide. A success-ful optimal MAPF solver is the conflict-based search (CBS) algorithm. CBS is a two level algorithm where special con-ditions ensure it returns the optimal solution. Solving MAPF optimally is proven to be NP-hard, hence CBS and all other optimal solvers do not scale up. We propose several ways to relax the optimality conditions of CBS trading solution qual-ity for runtime as well as bounded-suboptimal variants, where the returned solution is guaranteed to be within a constant fac-tor from optimal solution cost. Experimental results show the benefits of our new approach; a massive reduction in running time is presented while sacrificing a minor loss in solution quality. Our new algorithms outperform other existing algo-rithms in most of the cases.

Abstract In this paper, we consider the problem of planning collision-free trajec-tories to navigate a team of labeled robots from a set of start locations to a set of goal locations, where robots have pre-assigned and non-interchangeable goals. We present a solution to this problem for a centralized team operating in an obstacle-free, two-dimensional workspace. Our algorithm allows robots to follow Optimal Motion Plans (OMPs) to their goals when possible and has them enter Circular HOlding Patterns (CHOPs) to safely navigate congested areas. This OMP+CHOP algorithm is shown to be safe and complete, and simulation results show scalability to hundreds of robots. 1

We report a new method for computing near optimal makespan solutions to multi-robot path planning problem on graphs. Our focus here is with hard instances- those with up to 85 % of all graph nodes occupied by robots. Our method yields 100-1000x speedup compared with existing methods. At the same time, our solutions have much smaller and often optimal makespans. Introduction and Problem Formulation In this paper, we study centralized multi-robot path plan-ning problems on graphs, also known as cooperative path-finding (Silver 2005; Ryan 2008; Standley and Korf 2011; Surynek 2012b). Our focus is on finding plans with opti-

Abstract—We study the problem of path planning for unla-beled (indistinguishable) unit-disc robots in a planar environment cluttered with polygonal obstacles. We introduce an algorithm which minimizes the total path length, i.e., the sum of lengths of the individual paths. Our algorithm is guaranteed to find a solution if one exists, or report that none exists otherwise. It runs in time Õ m4 +m2n2, where m is the number of robots and n is the total complexity of the workspace. Moreover, the total length of the returned solution is at most OPT+4m, where OPT is the optimal solution cost. To the best of our knowledge this is the first algorithm for the problem that has such guarantees. The algorithm has been implemented in an exact manner and we present experimental results that attest to its efficiency. I.