Integer Feasibility of Random Polytopes

TL;DR

This paper investigates when random polytopes are likely to contain integer points, establishing a phase transition based on the inscribed ball radius, and introduces a randomized algorithm leveraging linear discrepancy for finding such points.

Contribution

It provides a probabilistic analysis of integer feasibility in random polytopes and links it to linear discrepancy, offering a constructive polynomial-time algorithm for integer point finding.

Findings

Random polytopes transition from infeasible to feasible with increasing inscribed ball radius.

A new connection between integer feasibility and linear discrepancy is established.

A randomized polynomial-time algorithm is developed for finding integer points in polytopes.

Abstract

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^O(sqrt{n}), there exist constants c_0 < c_1 such that with high probability, random polytopes are integer feasible if the radius of the largest ball contained in the polytope is at least c_1sqrt{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 c_0sqrt{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 (Lovett-Meka, FOCS '12) 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.