We provide an algorithm that maintains, against an adaptive adversary, a $(1-\varepsilon)$-approximate maximum matching in $n$-node $m$-edge general (not necessarily bipartite) undirected graph undergoing edge deletions with high probability with (amortized) $O(\mathrm{poly}(\varepsilon^{-1}, \log n))$ time per update. We also obtain the same update time for maintaining a fractional approximate weighted matching (and hence an approximation to the value of the maximum weight matching) and an integral approximate weighted matching in dense graphs. Our unweighted result improves upon the prior state-of-the-art which includes a $\mathrm{poly}(\log{n}) \cdot 2^{O(1/\varepsilon^2)}$ update time [Assadi-Bernstein-Dudeja 2022] and an $O(\sqrt{m} \varepsilon^{-2})$ update time [Gupta-Peng 2013], and our weighted result improves upon the $O(\sqrt{m}\varepsilon^{-O(1/\varepsilon)}\log{n})$ update time due to [Gupta-Peng 2013]. To obtain our results, we generalize a recent optimization approach to dynamic algorithms from [Jambulapati-Jin-Sidford-Tian 2022]. We show that repeatedly solving entropy-regularized optimization problems yields a lazy updating scheme for fractional decremental problems with a near-optimal number of updates. To apply this framework we develop optimization methods compatible with it and new dynamic rounding algorithms for the matching polytope.

翻译：我们提出了一种算法，该算法能够以高概率在经历边删除的$n$个节点、$m$条边的一般（不一定是二分）无向图中，针对自适应对手，维持一个$(1-\varepsilon)$-近似最大匹配，每次更新的（摊还）时间复杂度为$O(\mathrm{poly}(\varepsilon^{-1}, \log n))$。对于稠密图，我们还获得了相同的更新时间来维持一个分数近似加权匹配（从而近似最大权重匹配的值）以及一个整数近似加权匹配。我们的无权结果改进了先前的最新技术，其中包括$\mathrm{poly}(\log{n}) \cdot 2^{O(1/\varepsilon^2)}$的更新时间[Assadi-Bernstein-Dudeja 2022]和$O(\sqrt{m} \varepsilon^{-2})$的更新时间[Gupta-Peng 2013]；我们的加权结果改进了由于[Gupta-Peng 2013]导致的$O(\sqrt{m}\varepsilon^{-O(1/\varepsilon)}\log{n})$更新时间。为了获得这些结果，我们推广了[Jambulapati-Jin-Sidford-Tian 2022]中动态算法的一种近期优化方法。我们证明了，重复求解熵正则化优化问题可以为分数递减问题产生一种具有接近最优更新次数的惰性更新方案。为了应用此框架，我们开发了与之兼容的优化方法以及用于匹配多面体的新动态舍入算法。