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，输出结果在输入受到微小扰动时必须保持稳定，因为频繁的输出变化可能导致高昂的决策成本、用户信任度下降、潜在安全隐患以及可重复性缺失。本研究将算法的Lipschitz连续性作为稳定性度量，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嵌入方式的不同，可定义多种Lipschitz概念。我们主要关注在权重缩放下保持不变的一种概念，针对最小生成树问题、最短路径问题及最大权重匹配问题，分别提出了Lipschitz连续算法并给出了下界。特别地，我们首先针对无权重图设计了对抗边收缩的鲁棒算法，再将其应用于原始加权图所构造的无权重图，从而得到最短路径算法。随后，我们考虑另一种由输出解映射至其特征向量所诱导的Lipschitz概念。结果表明，针对该Lipschitz概念不存在Lipschitz连续算法，因此我们转而设计了具有有界逐点Lipschitz常数的算法，分别处理最小生成树问题和最大权重二分图匹配问题。其中，后者基于引入熵正则化的线性规划松弛方法。