crossing number inequality

In the previous lecture, we gave an upper bound on the second eigenvalue of the Laplacian of (bounded degree) planar graphs in order to analyze a simple spectral partitioning algorithm. A natural question is whether these bounds extend to more general families of graphs. Well-known generalizations of planar graphs are those which can be embedded on a surface of fixed genus, and, more generally, families of graphs that arise by forbidding minors. In fact, Spielman and Teng conjectured that for any graph excluding $K_h$ as a minor, one should have $\lambda_2 \lesssim \mathrm{poly}(h) d_{\max}/n$ . Of course planar graphs have genus 0, and by Wagner’s theorem, are precisely the graphs which exclude $K_5$ and $K_{3,3}$ as minors. In this lecture, we will follow an intrinsic approach of Biswal, myself, and Rao which, in particular, is able to resolve the conjecture of Spielman and Teng. First, we see why even pushing the conformal approach to bounded genus graphs is difficult.

Bounded genus graphs

For graphs of bounded genus, there is hope to use an approach based on conformal mappings. In 1980, Yang and Yau proved that

$\displaystyle \lambda_2(M) \lesssim \frac{g+1}{\mathrm{vol}(M)}$

for any compact Riemannian surface $M$ of genus $g$ . (Note that for the Laplace-Beltrami operator, one usually writes $\lambda_1$ as the first non-zero eigenvalue, rather than $\lambda_2$ .) In analog with Hersch’s proof of the genus 0 case, they use Riemann-Roch to obtain a degree- $(g+1)$ conformal mapping to the Riemann sphere, then try to pull back a second eigenfunction. A factor of the degree is lost in the Rayleigh quotient (hence the $g+1$ factor in the preceding bound), and Hersch’s Möbius trick is still required.


genus 0	genus 1	genus 2	genus 3

An analogous proof for graphs $G$ of bounded genus would proceed by constructing a circle packing of $G$ on the sphere $S^2$ , but instead of the circles having disjoint interiors, we would be assured that every point of $S^2$ is contained in at most $g$ circles. Unfortunately, such a result is impossible (this has to do with the handling of branch points in the discrete setting). Kelner has to take a different approach in his proof that $\lambda_2(G) \leq d_{\max}^{O(1)} (g+1)/n$ for graphs $G$ of genus at most $g$ .

He starts with a circle packing of $G$ on a compact surface $\mathbb S_0$ of genus $g$ (whose existence follows from results of Beardon and Stephenon and He and Schramm). Then Kelner randomly subdivides $G$ repeatedly, and these subdivisions give progressively better approximations to some sequence of surfaces $\{\mathbb S_i\}$ . Once the approximation is of high enough quality, one applies Riemann-Roch to $\mathbb S_k$ , and infers something about a subdivision of $G$ . The final element is to track how the second eigenvalue of $G$ changes (in expectation) under random subdivision.

Needless to say, this approach is already quite delicate, and for graphs that can’t be equipped with some kind of conformal structure, we seem to have reached a dead end. In this lecture, we’ll see how to use intrinsic deformations of the geometry of $G$ in order to bound its eigenvalues. Eventually, this will reduce to the study of certain kinds of multi-commodity flows.

Metrics on graphs

Let $G=(V,E)$ be an arbitrary n-vertex graph with maximum degree $d_{\max}$ . Recall that we can write

$\displaystyle \lambda_2 = \min_{f \neq 0 : \sum_{x \in V} f(x)=0} \frac{\sum_{xy \in E} |f(x)-f(y)|^2}{\sum_{x \in V} f(x)^2}.$

where $f : V \to \mathbb R$ . (Also recall that we can replace $\mathbb R$ by any Hilbert space, and the same formula holds.) The first step is to prepare this equality for “non-linearization” by getting rid of the linear condition $\sum_{x \in V} f(x)=0$ and the sum $\sum_{x \in V} f(x)^2$ . (This is a popular sort of passage in the non-linear geometry of Banach spaces, which also plays a rather important role in applications to the theoretical CS.) The goal is to get only terms that look like $|f(x)-f(y)|$ . Fortunately, there is a well-known way to do this:

$\displaystyle \lambda_2 = 2 n \cdot \min_{f : V \to \mathbb R} \frac{\sum_{xy \in E} |f(x)-f(y)|^2}{\sum_{x,y \in V} |f(x)-f(y)|^2},$

which follows easily from the equality $\sum_{x,y \in V} |f(x)-f(y)|^2 = 2n \sum_{x \in V} f(x)^2$ when $\sum_{x \in V} f(x)=0$ .

Thus if we want to bound $\lambda_2 = O(1/n)$ , we need to find an $f : V \to \mathbb R$ for which the latter ratio (without the $2n$ ) is $O(1/n^2)$ . Now, for someone who works a lot with linear programming relaxations, it’s very natural to consider a “relaxation”

$\displaystyle \gamma(G) = \min_{d} \frac{\sum_{xy \in E} d(x,y)^2}{\sum_{x,y \in V} d(x,y)^2},$

where the minimization is over all pseudo-metrics d, i.e. symmetric non-negative functions $d : V \times V \to \mathbb R$ which satisfy the triangle inequality, but might have $d(x,y)=0$ even for $x \neq y$ . Certainly $\gamma(G) \leq \lambda_2/2n$ , but Bourgain’s embedding theorem (which states that every n-point metric space embeds into a Hilbert space with distortion at most $O(\log n)$ ) also assures us that $\lambda_2(G) \leq O(n \log^2 n) \gamma(G)$ . Since we are trying to show that $\gamma(G) = O(1/n^2)$ , this $O(\log^2 n)$ term is morally negligible. One can see the paper for a more advanced embedding argument that doesn’t lose this factor, but for now we concentrate on proving that $\gamma(G) = O(1/n^2)$ . The embedding theorems allow us to concentrate on finding an intrinsic metric on the graph with small “Rayleigh quotient,” without having to worry about an eventual geometric representation.

As a brief preview… we are going to find a good metric $d$ by taking a certain kind of all-pairs multi-commodity flow at optimality, and weighting the edges by their congestion in the optimal flow. Thus as the flow spreads out on the graph, it has the effect of “uniformizing” its geometry.

Discrete Riemannian metrics, convexification, and duality

Let’s now assume that $G$ is planar. We want to show that $\gamma(G) = O(d_{\max}/n^2)$ . First, let’s restrict ourselves to vertex weighted metrics on $G$ . Given any non-negative weight function $\omega : V \to \mathbb R$ , we can define the length of a path $P$ in $G$ by summing the weights of vertices along it: $\mathsf{len}_{\omega}(P) = \sum_{x \in P} \omega(x)$ . Then we can define a vertex-weighted shortest-path pseudo-metric on $G$ in the natural way

$\displaystyle \mathsf{dist}_{\omega}(x,y) = \min \left\{ \mathsf{len}_{\omega}(P) : P \in \mathcal P_{xy}\right\},$

where $\mathcal P_{uv}$ is the set of all u-v paths in $G$ . We also have the nice relationship

$\displaystyle \sum_{xy \in E} \mathsf{dist}_{\omega}(x,y)^2 \leq 2 d_{\max} \sum_{x \in V} \omega(x)^2,\qquad(1)$

since $\mathsf{dist}_{\omega}(x,y) \leq [\omega(x)+\omega(y)]^2$ .

So if we define

$\displaystyle \Lambda_0(\omega) = \frac{\displaystyle \sum_{x \in V} \omega(x)^2}{\displaystyle \sum_{x,y \in V} \mathsf{dist}_{\omega}(x,y)^2}$

then by (1), we have $\gamma(G) \leq 2 d_{\max} \min_{\omega} \Lambda_0(\omega)$ .

Examples. Let’s try to exhibit weights $\omega$ for two well-known examples: the grid, and the complete binary tree.

For the $\sqrt{n} \times \sqrt{n}$ grid, we can simply take $\omega(x)=1$ for all $x \in V$ . Clearly $\sum_{x \in V} \omega(x)^2 = n$ . On the other hand, a random pair of points in the grid is $\Omega(\sqrt{n})$ apart, hence $\sum_{x,y \in V} \mathsf{dist}_{\omega}(x,y)^2 \approx n^2 \cdot (\sqrt{n})^2 = n^3$ . It follows that $\Lambda_0(\omega) = O(1/n^2)$ , as desired.

For the complete binary tree with root $r$ , we can simply put $\omega(r)=1$ and $\omega(x)=0$ for $x \neq r$ . (Astute readers will guess the geometrically decreasing weights are actually the optimal choice.) In this case, $\sum_{x \in V} \omega(x)^2 = 1$ , while all the pairs $x,y$ on opposite sides of the root have $\mathsf{dist}_{\omega}(x,y)=1$ . It again follows that $\Lambda_0(\omega) = O(1/n^2)$ . Our goal is to provide such a weight $\omega$ for any planar graph.