Lecture #6 discussion

A couple observations about Thursday’s lecture:

Most of you have probably seen Hall’s theorem before, which says that in a bipartite graph ${G = (U,V,E)}$, there is a matching ${M}$ that matches all the vertices on the left (i.e. has cardinality ${|U|}$) if and only if every set ${S \subseteq U}$ on the left has at least ${|S|}$ neighbors on the right. I wanted to point out that König’s theorem (which shows that the size of the maximum matching in ${G}$ is precisely ${vc(G)}$) is equivalent to Hall’s theorem. The proof is a fun exercise, so I’ll leave it out.

2. Polytope Integrality.

Something I mentioned in passing, but is worth saying slowly and more explicitly is this: several of the polyhedra defined by the LPs we considered in lecture were integral. (I.e., all the extreme points of these polyhedra belong to ${\mathbb{Z}^n}$, not just to ${\mathbb{R}^n}$.) Let’s go over these:

• When arguing about the min-(s,t)-cut LP, we said: consider any feasible solution ${x}$ to the min-cut LP, and any cost vector ${c}$. Then there exists an integer s-t cut ${(S_\alpha, \overline{S_\alpha})}$ with cost at most ${c^\top x}$. (The proof was via the randomized rounding.) Note that this s-t cut corresponds to an integer vector ${y \in R^{|E|}}$ where ${y_e = 1 \iff e \in S_\alpha \times \overline{S}_\alpha}$, which is also feasible for the cut LP.

This proof also gives us that the corresponding polyhedron ${K}$ is integral. To see this, consider any vertex ${x}$ of ${K}$. By the definition of vertex, there is some cost function such that ${x}$ is the unique minimizer for ${\min \{ c^\top x \mid x \in K\}}$. But since ${c^\top y \leq c^\top x}$, and ${y \in K}$, it follows that ${x = y}$ and hence ${x}$ is integral.

• For the bipartite matching LP, again consider any vertex ${x}$ of the polytope. We showed that if ${x}$ was fractional, it could be written as the average of two other solutions, and hence was not an extreme point. But since vertices and extreme points are the same, this contradicts the fact that ${x}$ was a vertex. Hence, this also shows that if we assign weights to the edges, an optimal BFS to this LP gives us the maximum weight matchings.
• An identical argument held for the bipartite vertex cover polyhedron, and hence for any assignment of weights to the vertices, an optimal BFS of this LP gives us a min-weight vertex cover.

We’ll revisit this issue of integral polytopes again in a few lectures.