  Real-world multi-agent systems such as warehouse robots operate under
significant time constraints -- in such settings, rather than spending
significant amounts of time solving for optimal paths, it is instead preferable
to find valid collision-free paths quickly, even if suboptimal, and given
additional time, to iteratively refine such paths to improve their cost. In
such domains, we observe that agent-agent collisions are sparse -- they involve
small local subsets of agents, and are geographically contained within a small
region of the overall space.
  Leveraging this insight, we can first plan paths for each agent individually,
and in the cases of collisions between agents, perform small local repairs
limited to local subspace windows. As time permits, these windows can be
successively grown and the repairs within them refined, thereby improving the
path quality, and eventually converging to the global joint optimal solution.
Using these insights, we present two algorithmic contributions: 1) the Windowed
Anytime Multiagent Planning Framework (WAMPF) for a class of anytime planners
that quickly generate valid paths with suboptimality estimates and generate
optimal paths given sufficient time, and 2) X*, an efficient WAMPF-based
planner. X* is able to efficiently find successive valid solutions by employing
re-use techniques during the repair growth step of WAMPF.
  Experimentally, we demonstrate that in sparse domains: 1) X* outperforms
state-of-the-art anytime or optimal MAPF solvers in time to valid path, 2) X*
is competitive with state-of-the-art anytime or optimal MAPF solvers in time to
optimal path, 3) X* quickly converges to very tight suboptimality bounds, and
4) X* is competitive with state-of-the-art suboptimal MAPF solvers in time to
valid path for small numbers of agents while providing much higher quality
paths.
