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.

翻译：组合算法广泛应用于决策制定与知识发现领域，确保其输出在输入发生微小扰动时保持稳定性至关重要。若未能实现这一目标，可能导致成本高昂的决策失误、用户信任度下降、潜在安全隐患以及结果不可复现等问题。遗憾的是，许多基础组合算法对输入扰动十分敏感。针对加权图问题中算法受输入扰动影响的现象，Kumabe与Yoshida（FOCS'23）近期提出了算法的Lipschitz连续性概念。本研究对该方法进行深入探索，针对最小顶点覆盖、集合覆盖及反馈顶点集等覆盖问题设计了Lipschitz连续算法。其中反馈顶点集问题的算法基于线性规划，在取整过程中我们创新性地开发并应用了名为"环稀疏化"的技术，该技术本身可能具有独立的研究价值。