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The approach to solving cooperative-path finding (CPF) as satisfiability (SAT) is revisited. An alternative encoding that exploits multi-valued state variables representing locations where a given agent resides is suggested. This encoding employs the ALL-DIFFERENT constraint to model the requirement that agents must not collide with each other. We show that our new domain-dependent encoding enables finding of optimal or near optimal solutions to CPFs in certain hard setups where A*-based techniques such as WHCA* fail to do so. Our finding is also that the ALL-DIFFERENT encoding can be solved faster than the existent encoding. Introduction and Context The problem of cooperative path-finding (CPF) (Silver, 2005; Ryan, 2008; Surynek, 2009) consists in finding noncolliding spatial-temporal paths for agents that need to relocate themselves from given initial locations to given goal locations. A generally adopted abstraction is that the environment is modeled as an undirected graph with agents placed in its vertices. At most one agent is placed in a vertex and at least one vertex remains unoccupied. The move is possible along an edge into a currently unoccupied vertex (an example instance of CPF on a 4-connected grid is shown in Figure 1). In our work we addressed the case of optimal or near optimal makespan and densely populated environments. We employ the SAT solving technology to optimize the makespan of solutions generated by existent fast suboptimal techniques such as BIBOX (Surynek, 2009) or PUSH-SWAP (Luna & Berkis, 2011). In contrast to the approach adopted in domain independent SAT-based planning (Kautz and Selman, 1999; Huang et al., 2010) we do not encode the whole problem as a SAT instance but only sub-problems represented by subsequences of the original solution are encoded. These (sub-optimal) sub-solutions are subsequently replaced by optimal ones found by the SAT solver. The similar approach has been recently apCopyright © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. This work is supported by the Czech Science Foundation (contract number GAP103/10/1287) and by the Japan Society for the Promotion of Science (contract number P11743). plied in domain-independent planning by Barták, Balyo, and Surynek (2012). We also propose a new compact domain dependent encodings for CPFs – called ALLDIFFERENT encoding – as an alternative to domain independent encodings used in SAT-based planning and to the encoding proposed in (Surynek, 2012). Cooperative Path-Finding (CPF) Formally An arbitrary undirected graph can be used to model the environment where agents are moving. Let be such a graph where and . The placement of agents in the environment is modeled by assigning them vertices of the graph. Let be a finite set of agents. Then, an arrangement of agents in vertices of graph will be fully described by a location function ; the interpretation is that an agent is located in a vertex . At most one agent can be located in each vertex; that is is uniquely invertible. Definition 1 (COOPERATIVE PATH FINDING). An instance of cooperative path-finding problem is a quadruple where location functions and define the initial and the goal arrangement of a set of agents in respectively. □ An arrangement at the -th time step can be transformed by a transition action which instantaneously moves agents in the non-colliding way to form a new arrangement . The resulting arrangement must satisfy the following validity conditions: (i) either or holds, (ii) , and (iii) . The task in cooperative path finding is to transform using above valid transitions to . 2 2