Optimal Multi-Agent Pathfinding Algorithms

The multi-agent path finding (MAPF) problem is a generalization of the single-agent path finding problem for k > 1 agents. It consists of a graph and a number of agents. For each agent, a unique start state and a unique goal state are given, and the task is to find paths for all agents from their start states to their goal states, under the constraint that agents cannot collide during their movements. In many cases there is an additional goal of minimizing a cumulative cost function such as the sum of the time steps required for every agent to reach its goal. The goal of my research is providing new methods to solve MAPF optimally and provide theoretical understandings that will help choose the best solver given a problem instance. Introduction and Research Question In the multi-agent path finding (MAPF) problem, we are given a graph, G(V,E), and a set of k > 1 agents labeled a1 . . . ak. Each agent ai has a start position si ∈ V and goal position gi ∈ V . At each time step an agent can either move to a neighboring location or can wait in its current location. The task is to return the least-cost set of actions for all agents that will move each of the agents to its goal without conflicting with other agents (i.e., without being in the same location at the same time or crossing the same edge simultaneously in opposite directions). In the online variant of this problem agents may appear at any time step and disappear once they reach their goal. MAPF has practical applications in video games, traffic control (Silver 2005; Dresner and Stone 2008), robotics (Bennewitz et al. 2002) and aviation (Pallottino et al. 2007). Up until now I presented two algorithms for solving MAPF optimally: the Increasing Cost Tree Search (Sharon et al. 2013b) (ICTS) and the Conflict Based Search (Sharon et al. 2012a) (CBS). ICTS is based on the understanding that a complete solution for the entire problem is built from individual paths (one for each agent). ICTS divides the MAPF problem into two levels. High level: At its high level, ICTS searches the increasing cost tree (ICT). Every node in the ICT consists of a k-ary vector [C1, . . . Ck] which represents all possible solutions in which the individual path cost of agent ai is exactly Ci. The root of the ICT is [opt1, ..., optk], where opti Copyright c © 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. is the optimal individual path cost for agent ai, i.e. the shortest path length from si to gi while ignoring other agents. A child in the ICT is generated by increasing the cost limit for one of the agents by one (or some unit cost). An ICT node [C1, .., Ck] is a goal if there is a complete non-conflicting solution such that the cost of the individual path for ai is exactly Ci. Low level: The low level acts as a goal test for the high level. For each ICT node [C1, .., Ck] visited by the high level, the low level is invoked. The task of the low level is to find a non-conflicting complete solution such that the cost of the individual path of agent ai is exactly Ci. For each agent ai, ICTS stores all single-agent paths of cost Ci in a special compact data-structure called a multi-value decision diagram (MDD) (Srinivasan et al. 1990). The low level searches the cross product of the MDDs in order to find a set of k non-conflicting paths for the different agents. If such a non-conflicting set of paths exists, the low level returns true and the search halts. Otherwise, false is returned and the high level continues to the next high-level node (of a different cost combination). 1. A set of constraints (N.constraints). Each constraint prohibits a given agent from being at a given coordinate (location, time step). The root of the CT contains an empty set of constraints. The child of a node in the CT inherits the constraints of the parent and adds one new constraint for one agent. 2. A solution (N.solution). A set of k paths, one path for each agent. The path for agent ai must be consistent with the constraints of ai. Such paths are found by the lowlevel search algorithm. 3. The total cost (N.cost). The cost of the current solution (summation over all the single-agent path costs). Node N in the CT is a goal node when N.solution is valid, i.e., the set of paths for all agents have no conflicts. The high-level phase performs a best-first search on the CT where nodes are ordered by their costs. Given the list of constraints for a node N of the CT, the low-level search is invoked. This search returns one shortest path for each agent, ai, that is consistent with all the constraints associated with ai in node N . If the chosen paths for two agents, a1 and a2, are found to conflict at coordinate x, two new CT nodes Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence