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\begin{problem}{(Final Project Next Step)} You will have received comments on your initial project ideas from problem set 2.  Using these comments, and rereading the ``Final Project Guidelines'' document on the course website  (\url{https://salil.seas.harvard.edu/classes/spectral-graph-theory}), refine your topic ideas and, if you wish, seek out 1-2 collaborators on Piazza.  Your group should submit
a project proposal (or two) approximately a half-page long.  (All group members should submit the same proposal.)
Include at least three citations to relevant background reading or related research literature, and be sure to indicate the nature of the project (e.g. synthesizing material that we will not cover in class, implementing and running experiments with algorithms using spectral graph theory, trying to obtain some small new theoretical results, etc.)  Try to anticipate any challenges you might encounter, and ask concrete questions where you could use additional pointers or guidance from us.
\end{problem}





\begin{problem}{(expander decompositions)}
In this problem, you'll see how to use Fiedler's Algorithm to partition an arbitrary graph into reasonably good expanders (as measured by $\nu_2$, the second smallest eigenvalue of the normalized Laplacian), while cutting few edges.  Efficient implementations of expander decompositions have found many algorithmic applications in recent years; intuitively they allow us to reduce solving a problem on an arbitrary graph to solving it on expanders, which is often an easier task.

Specifically, consider the following algorithm on input an unweighted, undirected graph $G=(V,E)$ and a parameter $\nu>0$:
\begin{enumerate}
    \item Initialize $E'=E$ and $G'=(V,E')$.
    \item While $G'=(V,E')$ contains a connected component $G_C=(C,E_C)$ such that $\nu_2(G_C) < \nu$, do:
    \begin{enumerate}
        \item Use Fiedler's Algorithm to find a cut $(S,C-S)$ of small conductance in $G_C$.
        \item Remove the edges between $S$ and $C-S$ from $E'$. 
    \end{enumerate}
\end{enumerate}
Observe that the above algorithm can be implemented in polynomial time, and outputs a final graph $G'=(V,E')$ in which every connected component $G_C$ has $\nu_2(G_C)\geq \nu$.   

Show that for any sufficiently small $\eps>0$ and an appropriate setting of $\nu=\Theta(\eps/\log m)^2$, the above algorithm will guarantee that $|E'|\geq (1-\eps)\cdot m$.  In particular, we can cut only an $\eps=1/\polylog(n)$ fraction of the edges and ensure that on every component, the lazy random walk mixes in time at most $\polylog(n)$.  (Hint: in the analysis, start the algorithm by assigning each edge a charge of 1, and whenever removing edges in a cut $(S,C-S)$, redistribute the charges of the cut edges uniformly over the edges of $S$ or $C-S$, whichever has fewer edges, to maintain a total charge of $m$.   Argue that each time the charge of an edge is increased, it increases by a multiplicative factor of at most $1+O(\sqrt{\nu})$.)
\end{problem}

\begin{problem}{(pseudoinverses)}
Let $A$ be an $n\times n$ complex matrix, with a singular value decomposition $A=U\Sigma V^*$ for unitary matrices $U$ and $V$, a diagonal, nonnegative real matrix $\Sigma$.  The {\em pseudoinverse} of $A$ is the matrix $A^+ = V\Sigma^+ U^*$, where $\Sigma^+$ is the diagonal matrix obtained by replacing each diagonal entry $\sigma$ of $\Sigma$ with 
$$\sigma^+ = \begin{cases} 0 & \text{if $\sigma=0$}\\ 1/\sigma & \text{otherwise} \end{cases}.$$
(It can be shown that $A^+$ does not depend on the choice of singular-value decomposition of $A$.)

\begin{enumerate}
    \item Prove that if the linear system $Ax=b$ has a solution, then $x=A^+b$ is a solution.
    \item Show that if $A$ is a Hermitian matrix, then $A^+$ is the unique matrix $B$ with the following property: for every eigenvector $v$ of $A$ of eigenvalue $\lambda$, $v$ is an eigenvector of $B$ of eigenvalue $\lambda^+$.  \label{part:reciprocal}
    \item Let $G$ be a connected, undirected graph with normalized Laplacian $N$.  We saw when discussing the power method that it is useful to have fast algorithms for applying $N^+$ to a vector.  Show that it suffices to be able to do this for regular graphs.  Specifically, let $\Greg$ be obtained by adding appropriately weighted self-loops to $G$ to make it regular and let $\Nreg$ be its normalized Laplacian.  Show that there are diagonal matrices $D_1$ and $D_2$ such that $N^+ = D_1 \Nreg^+ D_2$.  (Hint: relate $N^+$ to $L^+$ and observe the effect of adding self-loops on $L$.)
    \item Let $G$ be a connected, undirected, regular graph with normalized Laplacian $N=I-W$.  Prove that 
    $$N^+ = (I-W)^+ = \frac{1}{2} \left(I-J + (I+W)(I-W^2)^+(I+W) \right),$$
    where $J$ is the matrix in which every entry is $1/n$.
    Thus computing or applying the pseudoinverse of $I-W$ reduces to applying the pseudoinverse of $I-W^2$.  This recursion will be the basis of a nearly linear time algorithm that we see later in the course.  
\end{enumerate}
\end{problem}


\pagebreak
\begin{problem}{(Cayley expanders)}
\begin{enumerate}
\item For a prime $p$ and $k\in \N$, consider the group $\Z_p^k$, whose elements are $k$-dimensional vectors $a=(a_0,\ldots,a_{k-1})$ where each $a_i\in \{0,1,\ldots,p-1\}$ and the arithmetic is coordinate-wise addition modulo $p$: 
    $$(a_0,\ldots,a_{k-1})+(b_0,\ldots,b_{k-1}) = (a_0+b_0 \bmod p,\ldots,a_{k-1}+b_{k-1} \bmod p).$$
    Consider the following set $S\subseteq \Z_p^k$:
    $$S = \{(\alpha,\alpha\beta,\alpha\beta^2,\ldots,\alpha\beta^{k-1}) : \alpha,\beta\in \Z_p\},$$
    where again all arithmetic is modulo $p$.

Prove that the graph $G=\Cay(\Z_p^k,S)$ has $\omega(G)\leq (k-1)/p$.  

(Hint: you may use the facts that (a) a nonzero polynomial over $\Z_p$ of degree at most $k-1$ has at most $k-1$ roots in $\Z_p$\footnote{That is, for $f(x) = c_0+c_1x+c_2x^2+\cdots+c_{k-1}x^{k-1}$ where each $c_i\in \Z_p$ and not all $c_i$'s are zero, there are at most $k-1$ values $\beta\in \Z_p$ such that $f(\beta) \bmod p=0$).} and (b) if $\gamma$ is a nonzero element of $\Z_p$, then $\{\alpha\gamma : \alpha\in \Z_p\} = \Z_p$.)

Conclude that for infinitely many $n$ there is an explicit graph on $n$ vertices with spectral expansion at least $1/2$ and degree at most $O(\log^2 n).$\footnote{Those familiar with algebraic error-correcting codes may note that this construction works over any field, not just prime fields, and when we use a field of characteristic 2, the Cayley expander we obtain amounts to applying the correspondence between linear codes and Cayley graphs from Problem Set 1 to the code obtained by concatenating a Reed--Solomon code and a Hadamard code.}
\item  (extra credit) Let $\Gamma$ be an abelian group of size $n$, and let $G=\Cay(\Gamma,S)$ for $|S|=d$.  Prove that the diameter of $G$ is at least $n^{1/d}-1$ (or something similar, you do not need to get this exact expression).
(Hint: use commutativity!)
Conclude that if $G$ has spectral expansion $\gamma$ for a constant $\gamma>0$, then $$d=\Omega(\log n/\log\log n).$$  Thus abelian groups cannot yield Cayley expanders of constant degree.
\end{enumerate}
\end{problem}



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