Smoothed analysis of integer programming

Heiko Röglin, Berthold Vöcking

Abstract: We present a probabilistic analysis of integer linear programs (ILPs). More specifically, we study ILPs in a so-called smoothed analysis in which it is assumed that first an adversary specifies the coefficients of an integer program and then (some of) these coefficients are randomly perturbed, e.g., using a Gaussian or a uniform distribution with small standard deviation. In this probabilistic model, we investigate structural properties of ILPs and apply them to the analysis of algorithms. For example, we prove a lower bound on the slack of the optimal solution. As a result of our analysis, we are able to specify the smoothed complexity of classes of ILPs in terms of their worst case complexity. This way, we obtain polynomial smoothed complexity for packing and covering problems with any fixed number of constraints. Previous results of this kind were restricted to the case of binary programs.