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 encoding @cite_4 again employs log-space representation of variables but position of agent @math at time step @math , that is, @math is represented instead of representing vertex occupancy - that is, variables @math are represented using log-space encoding. To ensure that conflicts among agents in vertices do not occur, the constraint @cite_15 is intorduced for @math variables over all agents for each timestep @math . The advantage of the encoding is that various efficient encodings of the @cite_24 @cite_21 constraint over bit vectors can be integrated.
@cite_24 @cite_15 @cite_21 @cite_4
2142563857 2164279585 2398115889
This paper shows how all different constraints (ADCs) over bit-vectors can be handled within a SAT solver. It also contains encouraging experimental results in applying this technique to encode simple path constraints in bounded model checking. Finally, we present a new compact encoding of equalities and inequalities over bit-vectors in CNF. Many real-life Constraint Satisfaction Problems (CSPs) involve some constraints similar to the alldifferent constraints. These constraints are called constraints of difference. They are defined on a subset of variables by a set of tuples for which the values occuring in the same tuple are all different. In this paper, a new filtering algorithm for these constraints is presented. It achieves the generalized arc-consistency condition for these non-binary constraints. It is based on matching theory and its complexity is low. In fact, for a constraint defined on a subset of p variables having domains of cardinality at most d, its space complexity is O(pd) and its time complexity is O(p2d2). This filtering algorithm has been successfully used in the system RESYN ( 1992), to solve the subgraph isomorphism problem. A novel eager encoding of the ALL DIFFERENT constraint over bit-vectors is presented in this short paper. It is based on 1-to-1 mapping of the input bit-vectors to a linearly ordered set of auxiliary bit-vectors. Experiments with four SAT solvers showed that the new encoding can be solved order of magnitudes faster than the standard encoding in a hard unsatisfiable case.