Integer feasibility of random polytopes: random integer programs

We study the Chance-Constrained Integer Feasibility Problem, where the goal is to determine whether the random polytope P(A,b)={x ϵ Rn : Aix ≤ bi, i ϵ [m]} obtained by choosing the constraint matrix A and vector b from a known distribution is integer feasible with probability at least 1-ε. We consider the case when the entries of the constraint matrix A are i.i.d. Gaussian (equivalently are i.i.d. from any spherically symmetric distribution). The radius of the largest inscribed ball is closely related to the existence of integer points in the polytope. We find that for m up to 2O(√n) constraints (rows of A), there exist constants c0 < c1 such that with high probability (ɛ = 1 /poly(n)), random polytopes are integer feasible if the radius of the largest ball contained in the polytope is at least c1√log(m/n)); and integer infeasible if the largest ball contained in the polytope is centered at (1/2,...,1/2) and has radius at most c0√log(m/n)). Thus, random polytopes transition from having no integer points to being integer feasible within a constant factor increase in the radius of the largest inscribed ball. Integer feasibility is based on a randomized polynomial-time algorithm for finding an integer point in the polytope. Our main tool is a simple new connection between integer feasibility and linear discrepancy. We extend a recent algorithm for finding low-discrepancy solutions to give a constructive upper bound on the linear discrepancy of random Gaussian matrices. By our connection between discrepancy and integer feasibility, this upper bound on linear discrepancy translates to the radius bound that guarantees integer feasibility of random polytopes.