Lecture 15 notes

Some notes about today's lecture:

1. I mentioned that there are integrality results for some natural LPs for MST, branchings, shortest paths, matchings. Basically, this means that one can write LPs for these problems such that the basic solutions (i.e., vertices of the resulting polytope) are all 0-1 vectors. It was all a bit too fast towards the end, and so we will try to cover integrality for LPs in another lecture later.
   
   
2. For the branchings LP to make sense for negative edge weights, we should throw in a constraint that the total number of arcs chosen is at most n-1 (or that at most one arc should be chosen from the out-arcs of each non-root vertex). Else negative edge weights might cause the LP to choose too many edges, or even to choose edges to an extent greater than 1.
   
   
3. Finally, Srivatsan wondered about the fact that the size of the LP we wrote for branchings was exponentially sized: what's up with that? Two answers to this.
   
   One: we can use the LP purely for analytical purposes. If we use Edmonds' cycle shrinking algorithm from Lecture 5, we end up showing an integer primal solution of some value, and a feasible dual solution of the same value. And since these primal and dual solutions are feasible, and of equal value, both must be optimal!
   
   Second: wait until next lecture, we will show how to solve this exponentially sized LPs under certain conditions. For now, just observe that that if you are given a feasible (fractional) solution x, and it does not satisfy all the constraints, you can find some violating contraint in polynomial time. (In case there is a set S such that strictly less than 1 unit of x_e crosses it, there is a min-cut separating some vertex from the root with value strictly less than 1.) Note that if x is not in the branching polytope, this efficient procedure gives us a hyperplane that separates x from the polytope, and hence is called a "separation oracle" for this polytope.
   
   
4. Finally, I mentioned that Edmonds' result shows that every vertex of the polytope defined by the branching LP was integral. This is a slightly subtle point, so I'd like to elaborate.
   
   Edmonds shows that given edge-costs, the branching found by his algorithm wrt these edge costs is also an optimal LP solution for those edge-costs (since he gives a branching that is feasible for the primal, and a matching set of weights that are feasible for the dual).
   
   Now suppose the branching polytope has a fractional vertex (i.e., a extreme point z of this polytope such that not every coordinate of z is 0 or 1). One can show (using convexity etc) there exists some cost vector c for which this extreme point z is the unique optimal solution to the branching LP. But that would contradict the fact that for those edge costs, Edmonds' algorithm would find an integral optimal solution.

Next Monday, we'll give a quick overview of simplex, and then some details about the ellipsoid algorithm.