Graph algorithms are widely used for decision making and knowledge discovery. To ensure their effectiveness, it is essential that their output remains stable even when subjected to small perturbations to the input because frequent output changes can result in costly decisions, reduced user trust, potential security concerns, and lack of replicability. In this study, we consider the Lipschitz continuity of algorithms as a stability measure and initiate a systematic study of the Lipschitz continuity of algorithms for (weighted) graph problems. Depending on how we embed the output solution to a metric space, we can think of several Lipschitzness notions. We mainly consider the one that is invariant under scaling of weights, and we provide Lipschitz continuous algorithms and lower bounds for the minimum spanning tree problem, the shortest path problem, and the maximum weight matching problem. In particular, our shortest path algorithm is obtained by first designing an algorithm for unweighted graphs that are robust against edge contractions and then applying it to the unweighted graph constructed from the original weighted graph. Then, we consider another Lipschitzness notion induced by a natural mapping that maps the output solution to its characteristic vector. It turns out that no Lipschitz continuous algorithm exists for this Lipschitz notion, and we instead design algorithms with bounded pointwise Lipschitz constants for the minimum spanning tree problem and the maximum weight bipartite matching problem. Our algorithm for the latter problem is based on an LP relaxation with entropy regularization.

翻译：图算法广泛应用于决策制定与知识发现。为确保其有效性，算法的输出在输入发生微小扰动时需保持稳定——频繁的输出变化可能导致决策成本高昂、用户信任度下降、潜在安全隐患及结果不可复现。本研究以算法的Lipschitz连续性作为稳定性度量指标，系统性地探讨了（加权）图问题的Lipschitz连续性。根据输出解在度量空间中的嵌入方式，可定义多种Lipschitz性质。我们主要关注具有权重缩放不变性的Lipschitz性，针对最小生成树问题、最短路径问题及最大权匹配问题，给出了相应的Lipschitz连续算法与下界。其中，最短路径算法是通过先设计一种对边收缩具有鲁棒性的无向图算法，再将其应用于原始加权图构造的无权图而得到的。随后，我们考虑由输出解映射至其特征向量所诱导的另一种Lipschitz性质。结果表明，该Lipschitz概念下不存在Lipschitz连续算法，但我们针对最小生成树问题与最大权二分图匹配问题设计了具有有界逐点Lipschitz常数的算法。针对后者提出的算法基于带熵正则化的线性规划松弛。