  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.
