$\newcommand{\eps}{\varepsilon}$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) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time time used is $O(m\eps^{-1}\log(\eps^{-1}))$, where $m$ is the sum of the number of initially existing and inserted edges.

翻译：$\newcommand{\eps}{\varepsilon}$我们提出一种利用乘法更新（而非常数权重更新）的拍卖算法，用于在具有$n$个顶点和$m$条边的二分图上计算$(1-\eps)$-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相匹配。该算法非常简洁，可扩展为一种动态数据结构，在以下场景中维护$(1-\eps)$-近似最大权匹配：（1）单侧顶点删除（连带边）；（2）向另一侧单侧插入顶点（连带按权重排序的边）。总时间复杂度为$O(m\eps^{-1}\log(\eps^{-1}))$，其中$m$为初始边与插入边的总数之和。