$\newcommand{\eps}{\varepsilon}$ We present an auction algorithm using multiplicative instead of constant weight updates to compute a $(1-\eps)$-approximate maximum weight matching (MWM) in a bipartite graph with $n$ vertices and $m$ edges in time $O(m\eps^{-1}\log(\eps^{-1}))$, matching the running time of the linear-time approximation algorithm of Duan and Pettie [JACM '14]. Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a $(1-\eps)$-approximate maximum weight matching under (1) edge deletions in amortized $O(\eps^{-1}\log(\eps^{-1}))$ time and (2) one-sided vertex insertions. If all edges incident to an inserted vertex are given in sorted weight the amortized time is $O(\eps^{-1}\log(\eps^{-1}))$ per inserted edge. If the inserted incident edges are not sorted, the amortized time per inserted edge increases by an additive term of $O(\log n)$. The fastest prior dynamic $(1-\eps)$-approximate algorithm in weighted graphs took time $O(\sqrt{m}\eps^{-1}\log (w_{max}))$ per updated edge, where the edge weights lie in the range $[1,w_{max}]$.

翻译：$\newcommand{\eps}{\varepsilon}$ 我们提出一种采用乘法而非常数权重更新的竞价算法，用于在包含 $n$ 个顶点和 $m$ 条边的二分图中计算 $(1-\eps)$ 近似最大权匹配（MWM），时间复杂度为 $O(m\eps^{-1}\log(\eps^{-1}))$，与 Duan 和 Pettie [JACM '14] 的线性时间近似算法运行时间相匹配。我们的算法极为简洁，并可扩展为动态数据结构，在以下条件下维护 $(1-\eps)$ 近似最大权匹配：(1) 边删除的均摊时间为 $O(\eps^{-1}\log(\eps^{-1}))$；(2) 单侧顶点插入：若插入顶点关联的边已按权重排序，则每条插入边的均摊时间为 $O(\eps^{-1}\log(\eps^{-1}))$；若未排序，则每条插入边的均摊时间增加 $O(\log n)$ 的附加项。此前加权图中最快的动态 $(1-\eps)$ 近似算法每条更新边的时间复杂度为 $O(\sqrt{m}\eps^{-1}\log (w_{max}))$，其中边权取值范围为 $[1,w_{max}]$。