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：我们提出一种算法，能够在对抗性自适应删除边操作下，以高概率在$n$节点$m$边的一般图（不一定是二分图）中维护一个$(1-\varepsilon)$-近似最大匹配，且每次更新的均摊时间为$O(\mathrm{poly}(\varepsilon^{-1}, \log n))$。针对稠密图，我们同样获得了维护分数近似加权匹配（进而得到最大权匹配值的近似）以及整数近似加权匹配的相同更新时间。我们的无权重结果改进了先前最优方法，包括更新时间为$\mathrm{poly}(\log{n}) \cdot 2^{O(1/\varepsilon^2)}$的算法[Assadi-Bernstein-Dudeja 2022]和更新时间为$O(\sqrt{m} \varepsilon^{-2})$的算法[Gupta-Peng 2013]，而权重结果则改进了[Gupta-Peng 2013]中更新时间为$O(\sqrt{m}\varepsilon^{-O(1/\varepsilon)}\log{n})$的算法。为实现这一成果，我们推广了[Jambulapati-Jin-Sidford-Tian 2022]中用于动态算法的近期优化方法。研究表明，重复求解熵正则化优化问题可生成一种惰性更新机制，适用于更新次数接近最优的分数递减问题。为应用这一框架，我们开发了与之兼容的优化方法，并提出了适用于匹配多面体的新型动态舍入算法。