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We solve large-scale mixed-integer linear programs (MILPs) via distributed asynchronous saddle point computation. This is motivated by the MILPs being able to model problems in multi-agent autonomy, e.g., task assignment problems and trajectory planning with collision avoidance constraints in multi-robot systems. To solve a MILP, we relax it with a nonlinear program approximation whose accuracy tightens as the number of agents increases relative to the number of coupled constraints. Next, we form an equivalent Lagrangian saddle point problem, and then regularize the Lagrangian in both the primal and dual spaces to create a regularized Lagrangian that is strongly-convex-strongly-concave. We then develop a parallelized algorithm to compute saddle points of the regularized Lagrangian. This algorithm partitions problems into blocks, which are either scalars or sub-vectors of the primal or dual decision variables, and it is shown to tolerate asynchrony in the computations and communications of primal and dual variables. Suboptimality bounds and convergence rates are presented for convergence to a saddle point. The suboptimality bound includes (i) the regularization error induced by regularizing the Lagrangian and (ii) the suboptimality gap between solutions to the original MILP and its relaxed form. Simulation results illustrate these theoretical developments in practice, and show that relaxation and regularization together have only a mild impact on the quality of solution obtained.