 Job assignment problem solve with Branch and Bound algorithm

I started doing Branch and Bound Algorithm for assignment problem in C++ and i can't find the right solution. First of all assignment problem example: Ok so each person can be assigned to one job, and the idea is to assign each job to one of the person so that all the jobs are done in the quickest way.

Here is my code so far:
Code:
#include "Matrix.h"
// Program to solve Job Assignment problem
// using Branch and Bound

#include <limits.h>
#include <vector>
#include <algorithm>

using namespace std;

template<class T>

NUM getCost(Matrix& matrix, size_t x, size_t y, vector<bool>& colUsed);

void run(Matrix& matrix, vector<size_t>& assignedJobs);

int main()
{
Matrix matrix;
matrix.setMatrix(N);
matrix.print();

vector<size_t> assignedJobs;

run(matrix, assignedJobs);
cout << assignedJobs;
/*
cout << "size:E " << v.size() << endl;
for (vector<NUM>::iterator i = v.begin(); i != v.end(); i++)
{
cout << *i << endl;
}
*/
return 0;
}

// remember to use x only LOCALLY!!!
NUM getCost(Matrix& matrix, size_t x, size_t y, vector<bool>& colUsed)
{
// pathCost
NUM pathCost = matrix.matrix[x++][y];

for (size_t col = 0; col < matrix.size(); col++)
{
if (!colUsed.at(col))
{
NUM min =
#if defined NUM_INT
INT_MAX;
#endif
#if defined NUM_DBL
DBL_MAX;
#endif
size_t row = x;
for (; row < matrix.size(); row++)
{
if (min > matrix.matrix[row][col])
{
min = matrix.matrix[row][col];
}
}
pathCost += min;
}

}
return pathCost;
}

void run(Matrix& matrix, vector<size_t>& assignedJobs)
{
// array of used columns
vector<bool> colUsed;
for (size_t i = 0; i < matrix.size(); i++)
{
colUsed.push_back(false);
}

for (size_t row = 0; row < matrix.size(); row++)
{
size_t col = 0;
// bombona
while (colUsed.at(col++)); col--;
// choose the best job for the current worker
vector<NUM> jobs;
// get all the job costs from which to choose the smallest
// row++
jobs.push_back(getCost(matrix, col, row, colUsed));
// iterator at the position of the smallest element of jobs
vector<NUM>::iterator i_min = min_element(jobs.begin(), jobs.end());
// index of the smallest element in jobs
size_t index = (size_t)distance(jobs.begin(), i_min);

colUsed.at(index) = true;
assignedJobs.push_back(index);
}
}
I know i might be out of track. I didn't use lower bound in the beginning, i'm actually a little confused how this algorithm exactly works. So even step by step walktrough through the algorithm would be helpful.