Pointwise Lipschitz Continuous Graph Algorithms via Proximal Gradient Trajectory Analysis (2405.08938v3)

Abstract: In many real-world applications, it is undesirable to drastically change the problem solution after a small perturbation in the input as unstable outputs can lead to costly transaction fees, privacy and security concerns, reduced user trust, and lack of replicability. Despite the widespread application of graph algorithms, many classical algorithms are not robust to small input disturbances. Towards addressing this issue, we study the Lipschitz continuity of graph algorithms, a notion of stability introduced by Kumabe and Yoshida [KY23, FOCS'23] and further studied in various settings [KY24, ICALP'24], [KY25, SODA'25]. We give a general unifying framework for analyzing and designing pointwise Lipschitz continuous graph algorithms. In addition to being more general, our techniques obtain better bounds than can be achieved through extensions of previous work. First, we consider a natural continuous relaxation of the underlying graph problem along with a regularized objective function. Then, we develop a novel analysis of the distance between optimal solutions of the convex programs under small perturbations of the weights. 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 and maximum ($b$-)matching. To complement our algorithms, we show the tightness of our framework for the case of minimum $s$-$t$ cut by establishing tight lower bounds.