  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.
