Today we saw several definitions and facts about polytopes, including proofs of the integrality of the perfect matching polytope for bipartite graphs. Some comments.
- We saw the definition of the perfect matching polytope for non-bipartite graphs (with exponentially many constraints), but did not prove it correct. Do give it a try yourself (or read over a proof via the basic-feasible solution approach which appears in my hand-written notes, or in these notes).
- We saw a definition of the -arborescence polytope (again with exponentially many constraints). As an exercise, show that is also an integral polytope via the vertex approach.
Here’s how to go about it. In lecture #2, we saw an algorithm that finds the min-cost arborescence for non-negative edge costs. First, change it slightly to find min-cost arborescence for general edge costs . Naturally, gives you an (integer) solution to the LP . No reason to believe it is optimal for the LP, since this is the best integer solution and there may exist cheaper fractional solutions.
Next, take the dual of this LP. If you show a set of values for the dual such that the dual value equals the primal value, then the integer solution you found is also an optimal solution to the LP. (How would you do it? Think of the prices we were defining at the end of Lecture 2.) Now the optimal solution being an integer solution is true for every set of edge costs. So the vertices of the polytope must be all integral.
If this is a little vague, no worries: we will see the corresponding proofs for bipartite perfect matchings in the next lecture.
- Both the above LPs have exponential size. But has an “equivalent” poly-sized representation using some extra variables. Here’s a sketch. Indeed, the bad constraints are the ones saying that for every set not containing the root, . This is the same as saying that the min-cut between any vertex and the root is at least , when viewing as arc capacities. By maxflow-mincut, the maxflow from each vertex to the root (using arc-capacities ) is at least . So replace the exponentially many constraints by other constraints that enforce this max-flow condition. This may requires polynomially many extra variables. But “projected down” onto the variables the polytope is the same. This is another instance of the “projection example” I was attempting to draw in class. These things are called “extended formulations”.
- Does have a similarly small extended formulation? Recently Thomas Rothvoss showed that there are no tricks like the ones above, settling a decades-old conjecture of Yannakakis. In particular, every polytope that projects down to (for the complete graph ) must have constraints. See his slides/video for more details. As we remarked, this is an unconditional lower bound.
- For an extreme case when extended formulations help, Goemans shows a polytope with facets and extreme points that arises from projecting down a polytope in dimensions using facets.
- Finally, the use of total unimodularity for showing integrality of polytopes is indeed due to Alan Hoffman and Joe Kruskal. Here is their paper titled Integral Boundary Points of Convex Polyhedra, along with historical comments by the authors. Dabeen reminded me at the end of lecture that Edmonds and Giles actually developed a closely related concept, the theory of total dual integrality.