  Combinatorial algorithms are widely used for decision-making and knowledge
discovery, and it is important to ensure that their output remains stable even
when subjected to small perturbations in the input. Failure to do so can lead
to several problems, including costly decisions, reduced user trust, potential
security concerns, and lack of replicability. Unfortunately, many fundamental
combinatorial algorithms are vulnerable to small input perturbations. To
address the impact of input perturbations on algorithms for weighted graph
problems, Kumabe and Yoshida (FOCS'23) recently introduced the concept of
Lipschitz continuity of algorithms. This work explores this approach and
designs Lipschitz continuous algorithms for covering problems, such as the
minimum vertex cover, set cover, and feedback vertex set problems.
  Our algorithm for the feedback vertex set problem is based on linear
programming, and in the rounding process, we develop and use a technique called
cycle sparsification, which may be of independent interest.
