# Computing an optimal integer assignment given an optimal LP-solution

I modeled an ILP where I have a set of outfits and a set of friends with $|F| \leq |O|$, all these friends should take one outfit with the lowest effort $w(f,o) \geq 0$, considering the fact that these outfits differ in size, body form, and adjustments. The solution should be like this:

$min \sum_{i}^{|O|}\sum_{j}^{|F|}x_{ij}w(f,o))$

with the next constraints:

$\sum_{i\epsilon O}^{} . x_{ij =1} ,\forall j\in F$

$\sum_{j\epsilon F}^{} . x_{ij \leq 1} ,\forall i\in O$

$x_{i,j} \in \left \{ 0,1 \right \}$

The relaxation to LP would be to put:

$0\leq x_{ij}\leq1$

Now, considering the fact that we don't have an integrality gap in this problem and for every fractional LP-solution, there exists an integral feasible solution with the same cost, how can we give give a polynomial-time algorithm that, from any given optimal LP-solution, computes such an optimal integer assignment.