$\newcommand{\eps}{\varepsilon}$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：$\newcommand{\eps}{\varepsilon}$我们提出了一种使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$m$条边的二分图中计算$(1-\eps)$-近似最大权重匹配（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。我们的算法非常简单，且可扩展为动态数据结构，能在以下条件下维持$(1-\eps)$-近似最大权重匹配：(1) 一侧顶点删除（连带关联边）；(2) 另一侧顶点插入（关联边按权重排序）。总耗时为$O(m\eps^{-1}\log(\eps^{-1}))$，其中$m$为初始存在边与插入边的总数之和。