In a very recent breakthrough, Behnezhad and Ghafari [arXiv'24] developed a novel fully dynamic randomized algorithm for maintaining a $(1-\epsilon)$-approximation of maximum matching with amortized update time potentially much better than the trivial $O(n)$ update time. The runtime of the BG algorithm is parameterized via the following graph theoretical concept: * For any $n$, define $ORS(n)$ -- standing for Ordered RS Graph -- to be the largest number of edge-disjoint matchings $M_1,\ldots,M_t$ of size $\Theta(n)$ in an $n$-vertex graph such that for every $i \in [t]$, $M_i$ is an induced matching in the subgraph $M_{i} \cup M_{i+1} \cup \ldots \cup M_t$. Then, for any fixed $\epsilon > 0$, the BG algorithm runs in \[ O\left( \sqrt{n^{1+O(\epsilon)} \cdot ORS(n)} \right) \] amortized update time with high probability, even against an adaptive adversary. $ORS(n)$ is a close variant of a more well-known quantity regarding RS graphs (which require every matching to be induced regardless of the ordering). It is currently only known that $n^{o(1)} \leqslant ORS(n) \leqslant n^{1-o(1)}$, and closing this gap appears to be a notoriously challenging problem. In this work, we further strengthen the result of Behnezhad and Ghafari and push it to limit to obtain a randomized algorithm with amortized update time of \[ n^{o(1)} \cdot ORS(n) \] with high probability, even against an adaptive adversary. In the limit, i.e., if current lower bounds for $ORS(n) = n^{o(1)}$ are almost optimal, our algorithm achieves an $n^{o(1)}$ update time for $(1-\epsilon)$-approximation of maximum matching, almost fully resolving this fundamental question. In its current stage also, this fully reduces the algorithmic problem of designing dynamic matching algorithms to a purely combinatorial problem of upper bounding $ORS(n)$ with no algorithmic considerations.

翻译：在最近的一项突破性工作中，Behnezhad与Ghafari [arXiv'24] 提出了一种新颖的完全动态随机化算法，用于维持最大匹配的 $(1-\epsilon)$-近似，其摊销更新时间可能显著优于平凡的 $O(n)$ 更新时间。BG算法的运行时间通过以下图论概念参数化：* 对于任意 $n$，定义 $ORS(n)$——代表有序RS图——为在一个 $n$ 顶点图中，规模为 $\Theta(n)$ 的边不相交匹配 $M_1,\ldots,M_t$ 的最大数量，使得对于每个 $i \in [t]$，$M_i$ 在子图 $M_{i} \cup M_{i+1} \cup \ldots \cup M_t$ 中是一个诱导匹配。那么，对于任意固定的 $\epsilon > 0$，BG算法以高概率在 \[ O\left( \sqrt{n^{1+O(\epsilon)} \cdot ORS(n)} \right) \] 的摊销更新时间内运行，即使面对自适应对手也是如此。$ORS(n)$ 是一个更广为人知的关于RS图（要求每个匹配无论顺序如何都是诱导匹配）的量的密切变体。目前仅知 $n^{o(1)} \leqslant ORS(n) \leqslant n^{1-o(1)}$，而缩小这一差距似乎是一个众所周知的难题。在本工作中，我们进一步强化了Behnezhad与Ghafari的结果，并将其推向极限，从而得到一个随机化算法，其以高概率在 \[ n^{o(1)} \cdot ORS(n) \] 的摊销更新时间内运行，即使面对自适应对手也是如此。在极限情况下，即如果当前 $ORS(n) = n^{o(1)}$ 的下界几乎是最优的，我们的算法实现了 $n^{o(1)}$ 的更新时间用于最大匹配的 $(1-\epsilon)$-近似，几乎完全解决了这一基本问题。即使在现阶段，这也将设计动态匹配算法的算法问题完全简化为一个纯粹的组合问题——即上界估计 $ORS(n)$，而无需考虑算法设计因素。