Algorithm 1. Kuhn–Munkres Algorithm

Input: A bipartite graph $\mathrm{G}=\left({\mathrm{X}}_S,E,{\mathrm{X}}_T\right)$, corresponding edge weights $𝓌\left(x_{Si},x_{Tj}\right)$

Output: the perfect matching M.

Step 1: Generate initial labeling $\ell$ and match in ${\mathrm{G}}_{\ell }$

Step 2: If $M$ perfect, stop. Otherwise, pick a free vertex $x_{Si}\in {\mathrm{X}}_S$. Set $\mathrm{S}=x_{Si}$,$\mathrm{T}=\varnothing$.

Step 3: If ${ℐ}_S\left(X\right)=\mathrm{T}$, update labels (forcing ${ℐ}_S\left(X\right)\ne \mathrm{T}$) with following Equations (6) and (7)

$${\mathsf{\alpha }}_{\ell }=\underset{s\in S,y\notin T}{\min }\left\{𝓫\left(x_{Si}\right)+𝓫\left(x_{Tj}\right)-𝓌\left(x_{Si},x_{Tj}\right)\right\}$$ (6)

$$\widehat{𝓫}=\{\begin{array}{l}𝓫\left(\mathrm{x}\right)-{\mathsf{\alpha }}_{\ell }, x\in S\\ 𝓫\left(\mathrm{x}\right)+{\mathsf{\alpha }}_{\ell },\hspace{1em}x\in T\\ 𝓫\left(\mathrm{x}\right),\hspace{1em}otherwise\end{array}$$ (7)

Step 4: If ${ℐ}_S\left(X\right)\ne \mathrm{T}$, choose $\mathrm{y}\in {ℐ}_S\left(X\right)-T$:

If $\mathrm{y}$ free, $𝓊-𝓎$ is augmenting path. Augment M and go to 2

If $\mathrm{y}$ matched, say to $𝓏$, extend alternating tree: $\mathrm{S}=\mathrm{S}{\cup }^{ }𝓏$, $\mathrm{T}=\mathrm{T}{\cup }^{ }\mathrm{y}$. Go to