Lecture 4: Conformal mappings, circle packings, and spectral geometry

In Lecture 2, we used spectral partitioning to rule out the existence of a strong parallel repetition theorem for unique games.  In practice, spectral methods are a very successful heuristic for graph partitioning, and in the present lecture we’ll see how to analyze these partitioning algorithms for some common families of graphs.

Balanced separators, eigenvalues, and Cheeger’s inequality

Lipton and Tarjan proved that every planar graph has a negligibly small set of nodes whose remval splits the graph into two roughly equal pieces.  More specifically, every n-node planar graph can be partitioned into three disjoint sets A,B,S such that there are no edges from A to B, the separator S has at most O(\sqrt{n}) nodes, and |A|,|B| \geq n/3.  This allows one to do all sorts of things, e.g. a simple divide-and-conquer algorithm gives a linear time (1+\epsilon)-approximation for the maximum independent set problem in such graphs, for any \epsilon > 0.

So there is a natural question of how well spectral methods do, for example, on planar graphs.  Spielman and Teng showed that for bounded-degree planar graphs, a simple recursive spectral algorithm recovers a partition V=A \cup B of the vertex set so that |E(A,B)| = O(\sqrt{n}).  In other words, for bounded-degree planar graphs, spectral methods recover the Lipton-Tarjan separator theorem!  This is proved by combining Cheeger’s inequality with their main theorem.

Theorem [Spielman-Teng]: Every n-node planar graph with maximum degree d_{\max} has \displaystyle \lambda_2(G) = O\left(\frac{d_{\max}}{n}\right), where \lambda_2(G) is the second eigenvalue of the combinatorial Laplacian on G.

Recall that we introduced the combinatorial Laplacian in Lecture 2.  If G=(V,E) is an arbitrary finite graph, in this lecture it will make more sense to think about the Laplacian \Delta as an operator on functions f : V \to \mathbb R given by

\displaystyle \Delta f(x) = \mathrm{deg}(x) f(x) - \sum_{y : xy \in E} f(y).

If we define the standard inner product \langle f,g\rangle = \sum_{x \in V} f(x)g(x), then one can easily check that for any such f, we have \langle f, \Delta f\rangle = \sum_{xy \in E} |f(x)-f(y)|^2.  In particular, this implies that \Delta is a positive semi-definite operator.  If we denote its eigenvalues by \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n, then it is also easy to check that \lambda_1 = 0, with corresponding eigenfunction f(x)=1 for every x\in V.

Thus by standard variational principles, we have

\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}.

Let us also define the Cheeger constant h_G.  For an arbitrary subset S \subseteq V, let

\displaystyle h(S) = \frac{|E(S, \bar S)|}{\min(|S|,|\bar S|)},

note that this definition varies from the h we defined in Lecture 2, because we will be discussing eigenfunctions without boundary conditions.  Now one defines h_G = \min_{S \subseteq V} h(S).

Finally, we have the version of Cheeger’s inequality (proved by Alon and Milman in the discrete setting) for graphs without boundary.

Cheeger’s inequality: If G=(V,E) is any graph with maximum degree d_{\max}, then

\displaystyle \lambda_2(G) \geq \frac{h(G)^2}{2d_{\max}}.

This follows fairly easy from the Dirichlet version of Cheeger’s inequality presented in Lecture 2.  Here’s a sketch:  Let f : V \to \mathbb R satisfy \Delta f = \lambda_2 f, and suppose, without loss of generality, that V_+ = \{ x : f(x) > 0 \} has |V_+| \geq n/2.  Define f_+(x)=f(x) for f(x) > 0 and f_+(x)=0 otherwise.  Then f_+|_B = 0 for B = V \setminus V_+, so we can plug f_+ into the Dirichlet version of Cheeger’s inequality with boundary conditions on B.  For the full analysis, see this note which essentially follows this approach.  By examining the proof, note that one can find a subset S \subseteq V with h(S) \leq \sqrt{2 d_{\max} \lambda_2} by a simple “sweep” algorithm:  Arrange the vertices V = \{v_1, v_2, \ldots, v_n\} so that f(v_1) \leq f(v_2) \leq \cdots \leq f(v_n), and output the best of the n-1 cuts \{v_1, \ldots, v_i\}, \{v_{i+1}, \ldots, v_n\}.

So using the eigenvalue theorem of Spielman and Teng, along with Cheeger’s inequality, we can find a set S \subseteq V with h(S) \lesssim \sqrt{d_{\max}/n}.  While this cut has the right Cheeger constant, it is not necessarily balanced (i.e. \min(S, \bar S) could be very small).  But one can apply this algorithm recursively, perhaps continually cutting small chunks off of the graph until a balanced cut is collected.  Refer to the Spielman-Teng paper for details.  A great open question is how one might use spectral information about G to recover a balanced cut immediately, without the need for recursion.

Conformal mappings and circle packings

Now we focus on proving the bound \lambda_2(G) \lesssim d_{\max}/n for any planar graph G.  A natural analog is to look at what happens for the Laplace-Beltrami operator for a Riemannian metric on the 2-sphere.  In fact, Hersch considered this problem almost 40 years ago and proved that \lambda_2(M) \lesssim 1/\mathrm{vol}(M), for any such Riemannian manifold M.  His approach was to first use the uniformization theorem to get a conformal mapping from M onto S^2, and then try to pull-back the standard second eigenfunctions on S^2 \subseteq \mathbb R^3 (which are just the three coordinate projections).  Since the Dirichlet energy is conformally invariant in dimension 2, this almost works, except that the pulled-back map might not be orthogonal to the constant function.  To fix this, he has to post-process the initial conformal mapping with an appropriate Möbius transformation.

Unaware of Hersch’s work, Spielman and Teng derived eigenvalue bounds for planar graphs using the discrete analog of this approach:  Circle packings replace conformal mappings, and one still has to show the existence of an appropriate post-processing Möbius transformation.

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