## Lecture #7 discussion

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 ${x_e}$ for ${e \in E}$, and the constraints:

$\displaystyle \begin{array}{rl} x(\partial v) = 1 \qquad \qquad & \forall v \in V \\ x(\partial S) \geq 1 \qquad \qquad & \forall S \subset V, |S| \text{ odd} \\ x \geq 0 \qquad \qquad & \end{array}$

Note that in the second set of constraints, we can just consider ${S}$ of size at least ${3}$ and at most ${|V|-3}$: if ${|S| = 1}$ or ${|V|-1}$, then the first set of constraints already implies ${x(\partial S) = 1}$. So just focus on

$\displaystyle \begin{array}{rl} x(\partial v) = 1 \qquad \qquad & \forall v \in V \\ x(\partial S) \geq 1 \qquad \qquad & \forall S \subset V, |S| \text{ odd }, 3 \leq |S| \leq |V|-3\\ x \geq 0 \qquad \qquad & \end{array}$

Let us call this perfect matching polytope ${PM}$. 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 ${G = (V,E)}$ (minimizing the sum of ${|V| + |E|}$, say), and some vertex solution ${x}$ that is not integral. Clearly, ${|V|}$ must be even, else it will not satisfy the odd set constraint for ${S = V}$. First, the claim is that ${G}$ cannot have a vertex of degree~${1}$, 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 ${|E| \geq |V|-1}$, and neither being a cycle nor having a degree-${1}$ vertex implies that ${|E| \neq |V|}$. So ${|E| > |V|}$.

Recall there are ${|E|}$ variables. So any vertex/BFS is defined by ${|E|}$ tight constraints. If any of these tight constraints are the non-negativity constraints ${x_e \geq 0}$, then we could drop that edge ${e}$ and get a smaller counterexample. And since at most ${|V|}$ 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 ${W \subseteq V}$ with ${|W| \geq 3}$: i.e.,

$\displaystyle x(\partial W) = 1$

Now consider the two graphs ${G/W}$ and ${G/\overline{W}}$ obtained by contracting ${W}$ and ${\overline{W}}$ to a single new vertex respectively, and removing the edges lying within the contracted set. Since both ${W}$ and ${\overline{W}}$ have at least ${3}$ vertices, both are smaller graphs.

Now ${x}$ naturally extends to feasible solutions ${y}$ and ${z}$ for these new graphs. E.g., to get ${y}$, set ${y_e = x_e}$ for all edges ${e \in E \setminus \binom{W}{2}}$. Note that if the set ${W}$ got contracted to new vertex ${\widehat{w}}$ in ${G/W}$, then the fact that ${x(\partial W) = 1}$ implies that ${y(\partial \widehat{w}) = 1}$, and hence ${y}$ is a feasible solution to the perfect matching polytope for graph ${G/W}$. Similarly, ${z}$ is a feasible solution to the perfect matching polytope for graph ${G/\overline{W}}$.

By minimality of ${G}$, it follows that the perfect matching LP is integral for both ${G/W}$ and ${G/\overline{W}}$: i.e., the vertices of the perfect matching polytope for these smaller graphs all correspond to perfect matchings. And that means that

$\displaystyle y = \sum_i \lambda_i \cdot \chi_{M_i},$

where ${\chi_{M_i}}$ is the natural vector representation of the perfect matching ${M_i}$ in ${G/W}$, for values ${\lambda_i \geq 0, \sum_i \lambda_i = 1}$. Also, ${\lambda_i}$‘s can be taken to be rational, since ${y}$ is rational, as are ${\chi_{M_i}}$. Similarly, we have a rational convex combination

$\displaystyle z = \sum_i \mu_i \cdot \chi_{N_i},$

where ${N_i}$ are perfect matchings in ${G/\overline{W}}$. Since ${\lambda_i, \mu_i}$ are rationals, we could have repeated the matchings and instead written

$\displaystyle \begin{array}{rl} y &= \frac1k \sum_i \chi_{M_i} \\ z &= \frac1k \sum_i \chi_{N_i} \end{array}$

Finally, we claim that we can combine these to get

$\displaystyle x = \frac1k \sum_i \chi_{O_i}$

where ${O_i}$‘s are perfect matchings in ${G}$. How? Well, focus on edge ${e = \{l,r\} \in \partial W}$, with ${l \in W}$. Note that ${y_e = z_e = x_e}$. If we look at ${k}$ matchings ${M_i}$ in the sum for ${y}$: exactly ${x_e}$ fraction of these matchings ${M_i}$ — that is, ${kx_e}$ matchings — contain ${e}$. Similarly, exactly ${kx_e}$ of the matchings ${N_i}$ in the sum for ${x}$ contain ${e}$. Now we can pair such matchings (which share a common edge in ${\partial W}$) up in the obvious way: apart from the edge ${e}$, such an ${M_i}$ contains edges only within ${\overline{W}}$ and matches up all the vertices in ${\overline{W}}$ except vertex ${r}$, and ${N_i}$ contains edges only within ${W}$ and matches up all the vertices in ${W \setminus \{l\}}$. And ${e}$ matches up ${\{l,r\}}$. Hence putting together these perfect matchings ${M_i}$ and ${N_i}$ in ${G/W}$ and ${G/\overline{W}}$ gets us a perfect matching ${O_i}$ for ${G}$.

So ${x}$ can be written as a convex combination of perfect matchings of ${G}$. Hence, for ${x}$ 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 ${G}$, not just the perfect matchings. Here is the polytope Edmonds defined.

$\displaystyle \begin{array}{rl} x(\partial v) \leq 1 \qquad \qquad & \forall v \in V \\ \textstyle \sum_{e \in \binom{S}{2}} x_e \leq \frac{|S| - 1}{2} \qquad \qquad & \forall S \subset V, |S| \text{ odd } \\ x \geq 0 \qquad \qquad & \end{array}$

Clearly, all matchings in ${G}$ 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.