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 pointwise Lipschitz continuity of graph algorithms, a notion of stability introduced by Kumabe and Yoshida [KY23, FOCS'23] and further studied in related settings [KY24, ICALP'24], [KY25, SODA'25], [GKY25, ESA'25]. Our main result is a linear programming (LP) based minimum $S$-$T$ cut algorithm with a provably optimal Lipschitz constant, as witnessed by an accompanying lower bound. As a direct corollary, we give the first dynamic minimum $S$-$T$ cut algorithm with non-trivial recourse bound. At the core of our techniques is a novel framework for analyzing the Lipschitz constant of regularized LP relaxations. Our framework crucially unlocks the use of weighted regularizers, which could not be analyzed through previous methods, and leads to polynomial improvements in the Lipschitz constant compared to what is achievable through previous techniques. To demonstrate the flexibility of our methods, we also design an LP-based $b$-matching algorithm that improves on the state-of-the-art [KY23] Lipschitz constant in certain input regimes when $b\equiv 1$. Moreover, our algorithm cleanly extends to the general case when $b\geq 1$, whereas [KY23] is specialized to the case of $b\equiv 1$.