Question

I have to design a branch-and bound algorithm that solves the optimal tour of a graph on the cartesian plane every time. I have been given the hint that identifying hopeless branches earlier in the runtime will compound into a program that runs “a hundred times faster”. I had the idea of assuming that the shortest edge connected to the starting/ending node will be either the first or last edge in the tour but a thin diamond shaped graph proves otherwise. Does any one have ideas for how to eliminate these hopeless branches or a reference that talks about this?

Basically, is there a better way to branch to subsets of solutions better than just lexicographically, eg. first branch is including and excluding edge a-b, second branch includes and excludes branch a-c

Answer 1

So somewhere in your branch-and-bound algorithm, you look at possible places to go, and then somehow keep track of them to do later.

To make this more efficient, you can do a couple things:

1. Write a better bound calculator. In other words, come up with an algorithm that determines the bound more accurately. This will result in less time spent on paths that turn out to be poor.
2. Instead of using a stack to keep track of things to do, use a queue. Instead of using a queue, use a priority queue (heap) ordered by bound, e.g. the things that seem best are put at the top of the heap, and the things that seem bad are put on the bottom.