Thresholds in random integer programs

Karthik Chandrasekaran gave a talk at TTI today on the feasibility of integer programs. Given a polytope defined by the inequalities A x \le b in dimension n, can we say whether the polytope contains an integer point? In general, the problem is NP-hard, but efficient algorithms are known for special sub-cases. The goal in this talk was to understand if random instances of the problem are also hard.

The first thing to figure out is what do we mean by a random instance? Consider a point x_0 and the sphere S of radius R around x_0. Now draw m vectors x_1, x_2, \ldots, x_m uniformly from the surface of the unit sphere, and consider the polytope defined by faces which are tangent to S at x_0 + x_i for i = 1, 2, \ldots, m. That is, the vector x_0 + x_i is the normal vector of that face. This defines a random polytope whose distribution depends on the parameters (n,m,x_0,R). The goal is to find how R scales with n and m such that with high probability, the polytope contains an integer point for all x_0.

If the radius R is too small, then this will be hard to do because guaranteeing that an integer point is in the interior for all x_0 becomes hard. If R = \sqrt{n}/2, then we will always have an integer point. What should the real scaling for R look like?

The simple form of the main result is that if n \le m \le 2^{\sqrt{n}} and R \ge c_1 \sqrt{log(m/n)}, with high probability the polytope will have an integer point for every x_0. Conversely, if R \le c_0 \sqrt{ log(m/n) }, then with high probability the polytope “centered” at x_0 = (1/2,1/2,\ldots,1/2) with not contain an integer point. Note that if the number of faces is linear in n, then a constant radius is sufficient. I’m trying to square that with the “infinite dimensional spheres have vanishing volume and expanding surface area” but I think the fact that the polytope is “pointy” means that L_1 geometry gives better intuition.

To prove these bounds on R, they make a connection to the discrepancy of random Gaussian matrices (which approximate the random unit vector row matrices). The paper is on arxiv for those who want to take a look.

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