*In which we analyze the zig-zag graph product.*

# CS294 Lecture 16: Zig-Zag Graph Product

*In which we give an explicit construction of expander graphs of polylogarithmic degree, state the properties of the **zig-zag product* of graphs, and provide an explicit construction of a family of constant-degree expanders using the zig-zag product and the polylogarithmic-degree construction.

A *family of expanders* is a family of graphs , , such that each graph is -regular, and the edge-expansion of each graph is at least , for an absolute constant independent of . Ideally, we would like to have such a construction for each , although it is usually enough for most applications that, for some constant and every , there is an for which the construction applies in the interval , or even the interval . We would also like the degree to be slowly growing in and, ideally, to be bounded above by an explicit constant. Today we will see a simple construction in which and a more complicated one in which .

# CS294 Lecture 15: Abelian Cayley graphs

*In which we show how to find the eigenvalues and eigenvectors of Cayley graphs of Abelian groups, we find tight examples for various results that we proved in earlier lectures, and, along the way, we develop the general theory of harmonic analysis which includes the Fourier transform of periodic functions of a real variable, the discrete Fourier transform of periodic functions of an integer variable, and the Walsh transform of Boolean functions.*

Earlier, we prove the Cheeger inequalities

and the fact that Fiedler’s algorithm, when given an eigenvector of , finds a cut such that . We will show that all such results are tight, up to constants, by proving that

- The dimension- hypercube has and , giving an infinite family of graphs for which , showing that the first Cheeger inequality is exactly tight.
- The -cycle has , and , giving an infinite family of graphs for which , showing that the second Cheeger inequality is tight up to a constant.
- There is an eigenvector of the 2nd eigenvalue of the hypercube , such that Fiedler’s algorithm, given such a vector, outputs a cut of expansion , showing that the analysis of the Fiedler’s algorithm is tight up to a constant.

In this lecture we will develop some theoretical machinery to find the eigenvalues and eigenvectors of *Cayley graphs of finite Abelian groups*, a class of graphs that includes the cycle and the hypercube, among several other interesting examples. This theory will also be useful later, as a starting point to talk about constructions of expanders.

For readers familiar with the Fourier analysis of Boolean functions, or the discrete Fourier analysis of functions , or the standard Fourier analysis of periodic real functions, this theory will give a more general, and hopefully interesting, way to look at what they already know.

# CS 294 Lecture 14: ARV Analysis, Part 3

*In which we complete the analysis of the ARV rounding algorithm*

We are finally going to complete the analysis of the Arora-Rao-Vazirani rounding algorithm, which rounds a Semidefinite Programming solution of a relaxation of sparsest cut into an actual cut, with an approximation ratio .

In previous lectures, we reduced the analysis of the algorithm to the following claim.

# CS294 Lecture 13: ARV Analysis, cont’d

*In which we continue the analysis of the ARV rounding algorithm*

We are continuing the analysis of the Arora-Rao-Vazirani rounding algorithm, which rounds a Semidefinite Programming solution of a relaxation of sparsest cut into an actual cut, with an approximation ratio .

In previous lectures, we reduced the analysis of the algorithm to the following claim.

# CS294 Lecture 12: ARV Analysis

*In which we begin the analysis of the ARV rounding algorithm*

We want to prove

Lemma 1 (ARV Main Lemma)Let be a negative-type metric over a set such that the points are contained in a unit ball and have constant average distance, that is,

- there is a vertex such that for every
Then there are sets such that

- ;
- for every and every ,
where the multiplicative factors hidden in the and notations depend only on .

In this lecture, we will show how to reduce the ARV Main Lemma to a statement of the following form: if is a set of vectors such that the metric in the ARV Main Lemma can be written as , and is a random Gaussian vectors, and if is such that with probability, there are disjoint pairs such that and , then . We will then prove such a statement in the next lecture.

# CS294 Lecture 11: ARV

*In which we introduce semi-definite programming and a semi-definite programming relaxation of sparsest cut, and we reduce its analysis to a key lemma that we will prove in the next lecture(s)*

# CS294 Lecture 10: Bourgain’s Theorem

*In which we prove Bourgain’s theorem.*

Today we prove the following theorem.

Theorem 1 (Bourgain)Let be a semimetric defined over a finite set . Then there exists a mapping such that, for every two elements ,

where is an absolute constant. Given , the mapping can be found with high probability in randomized polynomial time in .

Together with the results that we proved in the last lecture, this implies that an optimal solution to the Leighton-Rao relaxation can be rounded to an -approximate solution to the sparsest cut problem. This was the best known approximation algorithm for sparsest cut for 15 years, until the Arora-Rao-Vazirani algorithm, which will be our next topic.

# CS294 Lecture 9: The Sparsest Cut Problem

*In which we introduce the sparsest cut problem and the Leighton-Rao relaxation.*

**1. The Uniform Sparsest Cut problem, Edge Expansion and **

Let be an undirected graph with vertices.

We define the uniform sparsity of a cut as

(we will omit the subscript when clear from the context) and the uniform sparsest cut of a graph is

In -regular graphs, approximating the uniform sparsest cut is equivalent (up to a factor of 2 in the approximation) to approximating the edge expansion, because, for every cut , we have

and, noting that, for every, ,

we have, for every ,

and so

It will be instructive to see that, in -regular graphs, is a relaxation of , a fact that gives an alternative proof of the easy direction of Cheeger’s inequalities.

# CS294 Lecture 8: Spectral Algorithms Wrap-up

*In which we talk about even more generalizations of Cheeger’s inequalities, and we analyze the power method to find approximate eigenvectors, thus having a complete description of a polynomial-time approximation algorithm for sparsest cut*