In the multi-agent path finding problem (MAPF) we are given a set of agents each with respective start and goal positions. The task is to find paths for all agents while avoiding collisions aiming to minimize an objective function. Two such common objective functions is the sum-of-costs and the makespan. Many optimal solvers were introduced in the past decade - two prominent categories of solvers can be distinguished: search-based solvers and compilation-based solvers. Search-based solvers were developed and tested for the sum-of-costs objective while the most prominent compilation-based solvers that are built around Boolean satisfiability (SAT) were designed for the makespan objective. Very little was known on the performance and relevance of the compilation-based approach on the sum-of-costs objective. In this paper we show how to close the gap between these cost functions in the compilation-based approach. Moreover we study applicability of various techniques developed for search-based solvers in the compilation-based approach. A part of this paper introduces a SAT-solver that is directly aimed to solve the sum-of-costs objective function. Using both a lower bound on the sum-of-costs and an upper bound on the makespan, we are able to have a reasonable number of variables in our SAT encoding. We then further improve the encoding by borrowing ideas from ICTS, a search-based solver. Experimental evaluation on several domains show that there are many scenarios where our new SAT-based methods outperforms the best variants of previous sum-of-costs search solvers - the ICTS, CBS algorithms, and ICBS algorithms.
SAT encoding. The next development has been done in SAT encoding called that separates conflict rules in MAPF and agents transitions between time steps @cite_51 . Conflict rules are expressed over anonymized agents that are encoded by direct variables @math .
@cite_51
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This paper addresses make span optimal solving of cooperative path-finding problem (CPF) by translating it to propositional satisfiability (SAT). The task is to relocate set of agents to given goal positions so that they do not collide with each other. A novel SAT encoding of CPF is suggested. The novel encoding uses the concept of matching in a bipartite graph to separate spatial constraint of CPF from consideration of individual agents. The separation allowed reducing the size of encoding significantly. The conducted experimental evaluation shown that novel encoding can be solved faster than existing encodings for CPF and also that the SAT based methods dominates over A based methods in environment densely occupied by agents. I describe and study the 'support encoding' of binary constraint satisfaction problems (CSP's) into boolean satisfiability (SAT). This is based on work by Kasif in 1990, in which clauses encode support information, rather than the encoding of conflicts in the standard 'direct encoding'. This enables arc consistency in the original CSP to be established by propagation in the translated SAT instance, providing an optimal worst case time algorithm for arc consistency [14]. I prove that the DP algorithm in SAT applied to the suport encoding behaves exactly like the MAC algorithm in CSP's. Using the SAT solver Chaff, I show that the support encoding can be used effectively to solve hard instances of random binary CSP's, and more effectively than the direct encoding on hard instances. Finally, I show that the local search algorithm WalkSAT can perform many times better on the support encoding than on the direct encoding. This paper deals with solving cooperative path finding (CPF) problems in a makespan-optimal way. A feasible solution to the CPF problem lies in the moving of mobile agents where each agent has unique initial and goal positions. The abstraction adopted in CPF assumes that agents are discrete units that move over an undirected graph by traversing its edges. We focus specifically on makespan-optimal solutions to the CPF problem where the task is to generate solutions that are as short as possible in terms of the total number of time steps required for all agents to reach their goal positions. We demonstrate that reducing CPF to propositional satisfiability (SAT) represents a viable way to obtain makespan-optimal solutions. Several ways of encoding CPFs into propositional formulae are proposed and evaluated both theoretically and experimentally. Encodings based on the log and direct representations of decision variables are compared. The evaluation indicates that SAT-based solutions to CPF outperform the makespan-optimal versions of such search-based CPF solvers such as OD+ID, CBS, and ICTS in highly constrained scenarios (i.e., environments that are densely occupied by agents and where interactions among the agents are frequent). Moreover, the experiments clearly show that CPF encodings based on the direct representation of variables can be solved faster, although they are less space-efficient than log encodings. The synchronous chip-rate discrete-time CDMA channel with channel coding is analysed and the corresponding factor graph is presented. Iterative joint decoding can be derived in a simple and direct way by applying the sum-product algorithm to the factor graph. Since variables have degree 2, no computation takes place at the variable nodes. Computation at the code nodes is just soft-in soft-out (SISO) decoding, whose output is the extrinsic PMF of the coded symbols. Computation at the channel transition nodes is equivalent to MAP symbol-by-symbol multiuser detection, whose complexity is generally exponential in K. We show that several previously proposed low-complexity algorithms based on interference cancellation (IC) can be derived in a simple direct way by approximating the extrinsic PMF output by the SISO decoders either as a single mass-point PMF (hard decision) or as a Gaussian PDF with the same mean and variance (moment matching). Differently from all previously presented methods (derived from heuristic reasoning), we see clearly that extrinsic rather than a posteriori PMF should be fed back. This yields important advantages in terms of limiting achievable throughput.