$\newcommand{\eps}{\varepsilon}$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 $O(m\eps^{-1})$, beating the running time of the fastest known approximation algorithm of Duan and Pettie [JACM '14] that runs in $O(m\eps^{-1}\log \eps^{-1})$. 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 used is $O(m\eps^{-1})$, where $m$ is the sum of the number of initially existing and inserted edges.

翻译：$\newcommand{\eps}{\varepsilon}$我们提出一种采用乘法而非常数权值更新的拍卖算法，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$O(m\eps^{-1})$，超越了Duan和Pettie [JACM '14] 已知最快近似算法的运行时间$O(m\eps^{-1}\log \eps^{-1})$。该算法非常简单，并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（连带其关联边）；（2）另一侧顶点插入（其关联边按权值排序）。总时间复杂度为$O(m\eps^{-1})$，其中$m$为初始存在边与插入边数量之和。