The Distributions of Functions Related to Parametric Integer Optimization

Abstract

We consider the asymptotic distribution of the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. To this end, we create a framework for studying the asymptotic distribution of general functions related to integer optimization. While there has been a significant amount of research focused around the extreme values that these functions can attain, little is known about their typical values. Each of these functions is defined for a fixed constraint matrix and objective vector while the right hand sides are treated as input. We show that the typical values of these functions are smaller than the known worst case bounds by providing a spectrum of probability-like results that govern their overall asymptotic distributions.