# algorithms – Weighted edge cover and matching problems in bipartite graphs

I am interested in the following problem for its own sake. This is not a homework problem.

Given a bipartite graph $$G = (V = X cup Y, E)$$ and a weight function $$w: E rightarrow mathbb{R}_{+}$$, how does one reduce the problem of finding a minimum weight edge cover to that of finding an optimal weight perfect matching?

apparently provides an answer which clearly I am not following. When discussing an edge cover, please use the definition — that is, a subset $$C subseteq E$$ such that each $$v in V$$ is incident to at least one $$e in C$$.