  This paper introduces a novel compact mixed integer linear programming (MILP)
formulation and a discretization discovery-based solution approach for the
Vehicle Routing Problem with Time Windows (VRPTW). We aim to solve the
optimization problem efficiently by constraining the linear programming (LP)
solutions to use only flows corresponding to time and capacity-feasible routes
that are locally elementary (prohibiting cycles of customers localized in
space).
  We employ a discretization discovery algorithm to refine the LP relaxation
iteratively. This iterative process alternates between two steps: (1)
increasing time/capacity/elementarity enforcement to increase the LP objective,
albeit at the expense of increased complexity (more variables and constraints),
and (2) decreasing enforcement without decreasing the LP objective to reduce
complexity. This iterative approach ensures we produce an LP relaxation that
closely approximates the optimal MILP objective with minimal complexity,
facilitating an efficient solution via an off-the-shelf MILP solver.
  The effectiveness of our method is demonstrated through empirical evaluations
on classical VRPTW instances. We showcase the efficiency of solving the final
MILP and multiple iterations of LP relaxations, highlighting the decreased
integrality gap of the final LP relaxation. We believe that our approach holds
promise for addressing a wide range of routing problems within and beyond the
VRPTW domain.
