  In many real-world applications, it is prohibitively expensive to drastically
change the solution to a problem after a small perturbation in the environment.
Therefore, the stability of an algorithm is a very desirable property. In this
paper, we study the class of pointwise Lipschitz continuous 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 (measured in $\ell_1$ distance) over the $\ell_1$ perturbations of the
weights of the underlying graph.
  In this work, we give a general and simple framework for bounding the
Lipschitz constant of algorithms measured through the unweighted $\ell_1$
distance of their outputs. Our approach consists of three main steps. First, we
consider a natural continuous relaxation of the underlying graph problem by
adding a smooth and strongly convex regularizer to the objective function.
Then, we give upper bounds on the $\ell_1$ distance of 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, densest subgraph, maximum $b$-matching, and
packing integer programs. To complement our algorithms, we show the tightness
of our results for certain problems by establishing tight lower bounds.
