**1. Perfect Matchings **

The proof of integrality of the non-bipartite perfect matching polytope was a bit fast. Let me give some more details.

Recall that the LP is defined by the variables for , and the constraints:

Note that in the second set of constraints, we can just consider of size at least and at most : if or , then the first set of constraints already implies . So just focus on

Let us call this perfect matching polytope . We’ll call the first set of equalities the *vertex constraints*, and the second set the *odd-set inequalities*.

Clearly, every perfect matching is a feasible solution to this LP. What the next theorem says is perfect matchings are precisely the vertices of this LP. (We already saw this for bipartite graphs, in Lecture~3.)

Theorem 1 (Perfect Matching Theorem)

Every vertex of this LP is integral.

Suppose not, and suppose there exists graphs for which there is a fractional vertex. Consider a minimal counterexample (minimizing the sum of , say), and some vertex solution that is not integral. Clearly, must be even, else it will not satisfy the odd set constraint for . First, the claim is that cannot have a vertex of degree~, or be disconnected (else we’ll get a smaller counterexample) or be just an even cycle (where we know this LP is indeed integral). Being connected implies that , and neither being a cycle nor having a degree- vertex implies that . So .

Recall there are variables. So any vertex/BFS is defined by tight constraints. If any of these tight constraints are the non-negativity constraints , then we could drop that edge and get a smaller counterexample. And since at most tight constraints come from the vertex constraints. So at least one odd-set constraint should be tight. Say this tight odd-set constraint is for the odd set with : i.e.,

Now consider the two graphs and obtained by contracting and to a single new vertex respectively, and removing the edges lying within the contracted set. Since both and have at least vertices, both are smaller graphs.

Now naturally extends to feasible solutions and for these new graphs. E.g., to get , set for all edges . Note that if the set got contracted to new vertex in , then the fact that implies that , and hence is a feasible solution to the perfect matching polytope for graph . Similarly, is a feasible solution to the perfect matching polytope for graph .

By minimality of , it follows that the perfect matching LP is integral for both and : i.e., the vertices of the perfect matching polytope for these smaller graphs all correspond to perfect matchings. And that means that

where is the natural vector representation of the perfect matching in , for values . Also, ‘s can be taken to be rational, since is rational, as are . Similarly, we have a rational convex combination

where are perfect matchings in . Since are rationals, we could have repeated the matchings and instead written

Finally, we claim that we can combine these to get

where ‘s are perfect matchings in . How? Well, focus on edge , with . Note that . If we look at matchings in the sum for : exactly fraction of these matchings — that is, matchings — contain . Similarly, exactly of the matchings in the sum for contain . Now we can pair such matchings (which share a common edge in ) up in the obvious way: apart from the edge , such an contains edges only within and matches up all the vertices in except vertex , and contains edges only within and matches up all the vertices in . And matches up . Hence putting together these perfect matchings and in and gets us a perfect matching for .

So can be written as a convex combination of perfect matchings of . Hence, for to be an extreme point (vertex) itself, it must be itself a perfect matching, and integral. This gives us the contradiction.

** 1.1. Max-Weight Matchings **

We didn’t get to this, but suppose you want to write an LP whose vertices are precisely (integral) matchings in , not just the perfect matchings. Here is the polytope Edmonds defined.

Clearly, all matchings in are feasible for this LP. Moreover, one can use the Perfect Matching Theorem above to show that every vertex of this polytope is also integral.

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