Question

I’m implementing the Hungarian algorithm in a project. I managed to get it working until what is called step 4 on Wikipedia. I do manage to let the computer create enough zeroes so that the minimal amount of covering lines is the amount of rows/columns, but I’m stuck when it comes to actually assign the right agent to the right job. I see how I could assign myself, but that’s more trial and error – i.e., I do not see the systematic method which is of course essential for the computer to get it work.

Say we have this matrix in the end:

   a  b  c  d
0  30 0  0  0
1  0  35 5  0
2  60 5  0  0
3  0  50 35 40

The zeroes we have to take to have each agent assigned to a job are (a, 3), (b, 0), (c,2) and (d,1). What is the system behind chosing these ones? My code now picks (b, 0) first, and ignores row 0 and column b from now on. However, it then picks (a, 1), but with this value picked there is no assignment possible for row 3 anymore.

Any hints are appreciated.

Answer 1

Well, I did manage to solve it in the end. The method I used was to check whether there are any columns/rows with only one zero. In such case, that agent must use that job, and that column and row have to be ignored in the future. Then, do it again so as to get a job for every agent.

In my example, (b, 0) would be the first choice. After that we have:

   a  b  c  d
0  x  x  x  x
1  0  x  5  0
2  60 x  0  0
3  0  x  35 40

Using the method again, we can do (a, 3), etc. I’m not sure whether it has been proven that this is always correct, but it seems it is.