We study the classical weighted perfect matchings problem for bipartite graphs or sometimes referred to as the assignment problem, i.e., given a weighted bipartite graph $G = (U\cup V,E)$ with weights $w : E \rightarrow \mathcal{R}$ we are interested to find the maximum matching in $G$ with the minimum/maximum weight. In this work we present a new and arguably simpler analysis of one of the earliest techniques developed for solving the assignment problem, namely the auction algorithm. Using our analysis technique we present tighter and improved bounds on the runtime complexity for finding an approximate minumum weight perfect matching in $k$-left regular sparse bipartite graphs.

翻译：我们研究的是典型的两边图表的加权完美匹配问题,或有时称之为任务分配问题,即,考虑到加权双边图形$G=(U\cup V,E),重量为$w:E\rightrow \mathcal{R},我们有兴趣找到以美元为单位与最低/最大重量的最大匹配。在这项工作中,我们对为解决任务分配问题而开发的最早的技术之一,即拍卖算法,进行了新的、可以说更简单的分析。利用我们的分析技术,我们提出了更严格、更完善的运行时复杂度,以寻找以美元偏左的普通稀薄两边图表为单位的近似微米重量完美匹配。