The World’s Simplest Interesting Algorithm

In this post, I want to tell you about what I think might be the world’s simplest interesting algorithm.

The vertex cover problem. Given a graph , we want to find the smallest set of vertices such that every edge is covered by the set . This means that for each edge , at least one of or is in .

The vertex cover problem is known to be NP-complete, meaning that it is very hard (or impossible) to find a polynomial-time algorithm for the problem.

But what we can do is find an approximately optimal solution to the problem. There’s a beautiful and simple algorithm that finds a set whose size is guaranteed to be within a factor of of optimal.

The algorithm. To build our set , we repeatedly find some edge that is not yet covered, and we add both and to our set . That’s the entire algorithm.

At first sight, this seems like a crazy algorithm. Why would we put both and into , when we only need one of or in order to cover the edge ?

The answer to this question is simple. If we define to be the optimal solution to the vertex-cover problem, then must contain at least one of or . By placing both of and into our set , we guarantee that at least half the elements in our set also appear in . That means that, once our algorithm finishes, we are guaranteed that .

A challenge.  This might be the world’s simplest interesting algorithm (and analysis).  But it also might not be. What do you think?