Actually, you can solve the assignment problem as a linear program (LP). Just do

    min sum_i sum_j c_{ij}*x_{ij}

    s.t. sum_i x_{ij} == 1, for all j
         sum_j x_{ij} == 1, for all i

and you automagically get an integer solution in the x_{ij} due to the structure of the constraints.

CVXPY would be a good way to implement this these days.