Algorithm 1. Pseudocode for the Hungarian algorithm Function Hungarian-Algorithm ($G$).

% initial vertex feasible labeling $l_1$

$i=1,j=1$

$\forall x\in X,l_1(x)=\underset{x\in X}{\min }\{w(e_{xy})\}$

$\forall y\in Y,l_1(y)=0$

$M_1\leftarrow \mathrm{initial}\mathrm{matching}$

while $\left|M_i\right|<\left|X\right|$

% find the maximum matching $M$

$M_i^j=M_i$

do

$F_i^{j+1}\leftarrow \mathrm{alternating}\mathrm{forest}\mathrm{of}{M}_i^j\mathrm{rooted}\mathrm{at}\mathrm{the}\mathrm{unmatched}\mathrm{vertices}x\in X$

$P_i^{j+1}\leftarrow \mathrm{a}\mathrm{path}\mathrm{of}{F}_i^{j+1}\mathrm{which}\mathrm{contain}\mathrm{another}\mathrm{unmatched}\mathrm{vertex}y\in Y$

$M_i^{j+1}=M_i^j\oplus E(P_i^{j+1})$

$Q_{}^{j+1}=[X-V(F_i^{j+1})]\cup [Y\cap V(F_i^{j+1})]$

$j=j+1$

while $\left|M_i^j\right|\ne \left|Q_i^j\right|$

$M_i=M_i^j$

% update the feasible vertex labeling

$F_i\leftarrow \mathrm{alternating}\mathrm{forest}\mathrm{of}{M}_i\mathrm{rooted}\mathrm{at}\mathrm{the}\mathrm{unmatched}\mathrm{vertices}x\in X$

$S=V(F_i)\cap X$

$T=V(F_i)\cap Y$

$\alpha =mi{n}_{x\in S,y\notin T}\{w(e_{xy})-l_i(x)-l_i(y)\}$

$l_{i+1}(v)=\left\{\begin{array}{c}{l}_i(v)+\alpha \\ l_i(v)-\alpha \\ l_i(v)\end{array}\right.\begin{array}{c}\mathrm{if}v\in S\\ \mathrm{if}v\in T\\ \mathrm{otherwise}\end{array}$

$i=i+1$

done