In this paper, we consider the problem of maintaining a $(1-\varepsilon)$-approximate maximum weight matching in a dynamic graph $G$, while the adversary makes changes to the edges of the graph. In the fully dynamic setting, where both edge insertions and deletions are allowed, Gupta and Peng gave an algorithm for this problem with an update time of $\tilde{O}_{\varepsilon}(\sqrt{m})$. We study a natural relaxation of this problem, namely the decremental model, where the adversary is only allowed to delete edges. For the cardinality version of this problem in general (possibly, non-bipartite) graphs, Assadi, Bernstein, and Dudeja gave a decremental algorithm with update time $O_{\varepsilon}(\text{poly}(\log n))$. However, beating $\tilde{O}_{\varepsilon}(\sqrt{m})$ update time remained an open problem for the \emph{weighted} version in \emph{general graphs}. In this paper, we bridge the gap between unweighted and weighted general graphs for the decremental setting. We give a $O_{\varepsilon}(\text{poly}(\log n))$ update time algorithm that maintains a $(1-\varepsilon)$-approximate maximum weight matching under adversarial deletions. Like the decremental algorithm of Assadi, Bernstein, and Dudeja, our algorithm is randomized, but works against an adaptive adversary. It also matches the time bound for the cardinality version upto dependencies on $\varepsilon$ and a $\log R$ factor, where $R$ is the ratio between the maximum and minimum edge weight in $G$.

翻译：本文研究动态图 $G$ 中维护 $(1-\varepsilon)$ 近似最大权重匹配的问题，其中对手可对图的边进行修改。在全动态设置（允许边的插入与删除）下，Gupta 和 Peng 给出了更新时间为 $\tilde{O}_{\varepsilon}(\sqrt{m})$ 的算法。我们研究该问题的自然松弛模型——递减模型，其中对手仅允许删除边。对于一般（可能非二分）图中的基数版本问题，Assadi、Bernstein 和 Dudeja 给出了更新时间为 $O_{\varepsilon}(\text{poly}(\log n))$ 的递减算法。然而，在一般图的*加权*版本中，突破 $\tilde{O}_{\varepsilon}(\sqrt{m})$ 更新时间仍是开放问题。本文填补了递减设置下无加权图与一般加权图之间的空白：我们提出更新时间为 $O_{\varepsilon}(\text{poly}(\log n))$ 的算法，可在对抗性删除操作下维护 $(1-\varepsilon)$ 近似最大权重匹配。与 Assadi、Bernstein 和 Dudeja 的递减算法类似，本算法为随机算法，但可抵抗自适应对手。该算法在依赖 $\varepsilon$ 及 $\log R$ 因子（其中 $R$ 为图 $G$ 中最大与最小边权之比）的意义下，与基数版本的时间界相匹配。