Phase Transition in Random Integer Programs

We study integer programming instances over polytopes P (A, b) = {x : Ax ≤ b} where the constraint matrices A are random – the rows of the constraint matrices are chosen i.i.d. from a spherically symmetric distribution. We address the radius of the largest inscribed ball that guarantees integer feasibility of such random polytopes with high probability. We show that for m = 2 √ , there exist constants c0 < c1 such that with high probability, random polytopes are 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); and they are integer feasible for every center if the radius is at least c1 √ 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. A recent algorithm for finding low-discrepancy solutions [16] leads to a randomized polynomialtime algorithm for finding an integer point if random polytopes contain a ball of radius at least 16c1 √ log (m/n). ∗{karthe, vempala}@gatech.edu, School of Computer Science, Georgia Institute of Technology; the first author was supported in part by an Algorithms and Randomness Center (ARC) fellowship; both authors were supported in part by the NSF.