Justification for the Hungarian method (Kuhn-Munkres) I wrote an implementation of the Kuhn-Munkres algorithm for the minimum-weight bipartite perfect matching problem based on lecture notes I found here and there on the web. It works really well, even on thousands of vertices. And I agree that the theory behind it is truly beautiful. And yet I still wonder why I had to go to such lengths. I find that these lecture notes don't explain why we can't simply take the primal linear program and pass it to the simplex method. Of course I suspect that it's a question of predictable performance, but since I haven't seen it explicitly stated, I'm not too sure. The polytope's extreme points of the primal are proven to be in 0-1, so it seems that we can feed it directly to a simplex implementation without even formulating the dual. Or am I being simplistic?
(Moved from a comment.)

Of course you can solve any LP by using a general-purpose LP solver, but specialised algorithms typically have a much better performance.

It is not only about theoretical asymptotic performance guarantees, it is also about practical real-world performance. Algorithms such as the Hungarian method can be extremely streamlined and they are relatively easy to implement correctly and efficiently.

You can also often avoid issues like using exact rational numbers vs. floating point numbers; everything can be done easily with integers.