Integer Feasibility of Random Polytopes

We study integer programming instances over polytopes P (A, b) = {x : Ax ≤ b} where the constraint matrix A is random, i.e., its entries are i.i.d. Gaussian or, more generally, its rows are i.i.d. from a spherically symmetric distribution. The radius of the largest inscribed ball is closely related to the existence of integer points in the polytope. We show that for m = 2 √ , there exist constants c0 < c1 such that with high probability, 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. We show integer feasibility via 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 [15] to give a constructive upper bound on the linear discrepancy of random matrices. By our connection between discrepancy and integer feasibility, this upper bound on linear discrepancy translates to the radius lower bound that guarantees integer feasibility of random polytopes. ∗[email protected], School of Engineering and Applied Sciences, Harvard University; This work was done while the author was a student at Georgia Institute of Technology supported in part by the Algorithms and Randomness Center (ARC) fellowship and the NSF. †[email protected], School of Computer Science, Georgia Institute of Technology; Supported in part by the NSF. 1 ar X iv :1 11 1. 46 49 v3 [ cs .D S] 2 3 A ug 2 01 3