Bipartite graph is a special graph. Its vertices are divided into two separate sets and edges only exist between those two sets. The maximum cardinality matching problem of the bipartite graph can be modeled as a linear programming problem below:
max sum_{(i,j) in E} x_{ij}
s.t. sum_{j:(i,j) in E} x_{ij} le 1 forall i in I,
sum_{i:(i,j) in E} x_{ij} le 1 forall j in J,
x_{ij} ge 0 forall (i,j) in E.
in which I, J stand for the left and right sets and E stands for set of the edges. Write down duality of the above LP.