Lipschitz Continuous Graph Algorithms via Proximal Gradient Analysis

Abstract: We study the class of Lipschitz continuous graph algorithms as introduced in the recent work of Kumabe and Yoshida [FOCS'23]. The Lipschitz constant of an algorithm, intuitively, bounds the ratio of the changes in its output over the perturbations of its input. Our approach consists of three main steps. First, we consider a natural convex relaxation of the underlying graph problem with the addition of a smooth and strongly convex regularizer. Then, we give upper bounds on the ell-1 distance between the optimal solutions of the convex programs, under small perturbations of the weights, via a stability analysis of the trajectory of the proximal gradient method. Finally, we present new problem-specific rounding techniques to obtain integral solutions to several graph problems that approximately maintain the stability guarantees of the fractional solutions. We apply our framework to a number of problems including minimum s-t cut, multiway cut, densest subgraph, maximum b-matching, and packing integer programs. Finally, we show the tightness of our results for certain problems by establishing matching lower bounds.