Friday, 6 November 2015

Lecture 12

The slides I used today to talk about the rendezvous problem are linked here. The paper is

R. R. Weber, Optimal symmetric rendezvous search on three locations, Math Oper Res., 37(1): 111-122, 2012.

It was in solving this problem that I first came to study semidefinite programming. I am fond of this paper because of the way it employs a range of mathematical tools on the way to proving the main theorem. Indeed, I today discovered that there is an AMS review of this paper which says, "The author dedicates many pages to the proof of the theorem, utilizing multiple ideas from probability theory, game theory, linear algebra, linear programming, and semidefinite programming."

I mentioned today the class APX, consisting of optimization problems that are approximable, in the sense that there exists a polynomial time algorithm which can always find an answer that is within a fixed multiplicative factor of the optimal answer. One example of an APX problem is the travelling salesman problem with distances satisfying the triangle inequality. For this problem there exists a 2-approximation algorithm based on the efficient solution of the minimum spanning tree problem. (This will be a question for you on Examples sheet 2.) This problem is also APX-complete, in that it is as hard as any other in problem in APX.

Another APX-complete problem is minimum vertex cover. This also has a 2-approximation algorithm: just find a maximum matching and then include in the vertex cover both endpoints of the edges in the matching. Do you see why this works?

We now know about P, NP, NP-hard, NP-complete and APX. There is a huge "zoo" of complexity classes. The entertaining complexity zoo web site lists 498 complexity classes that have been defined by researchers.