Wednesday, 4 November 2015

Lecture 11

At the start of today's lecture I explained the Hungarian algorithm, which is a polynomial time algorithm for solving the assignment problem. My emphasis was on explaining why it works. In practice the algorithm looks rather different. The fact that one is adjusting node numbers and solving maximum matching problems is rather hidden from view.

You might like to read this nice tutorial on the Hungarian algorithm and read through the simple Example 2 on pages 18-27.

The algorithms in Lecture 11 are all polynomial time algorithms: Bellman-Ford, Dijkstra's and Prim's algorithms. They do involve numbers $c_{ij}$, but the the calculations that must be done with these numbers, such as adding (to find $c_{ij}+\lambda_j$) and comparing (to find $\min\{c_{it},c_{ij}+c_{jt}\}$) can all be done in time that is polynomial in the size of the input, as measured in numbers of binary bits. This is why running time is depends most crucially on the number of vertices $|V|$ and number of edges $|E|$.

One should be aware that for graphs with special structure, perhaps very sparse in edges, there exist specially tailored versions of some of our algorithms, which have better running time than those quoted in this lecture for a complete graph where $|E|=|V|^2$.