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\lecture{22}{May 2, 2007}{Madhu Sudan}{Jelani Nelson}
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\section{Overview}
This lecture describes a natural DNP-complete problem first proposed
by Impagliazzo and Levin \cite{IL90}.
\section{Universal Problems for DNP}
We begin by recalling the definition of the complexity class DNP
(Distributional NP) from last lecture.
\begin{definition}
{\em DNP} is the class of languages $(\pi, D)$ where $\pi$ is an
efficiently
computable function specifying an NP language (given an input $x$
find a poly-size witness $y$ such that $\pi(x, y) = 1$), and $D$ is
a poly-time sampleable distribution. That is, there is a uniform
poly-time algorithm $G$ such that $G(\{0,1\}^n)\subseteq \{0,1\}^n$,
and an input $x$ of length $n$ is drawn according to the
distribution $G(U_n)$, i.e. if $D_n$ is the distribution on
length-$n$ inputs specified by $D$ then $D_n = G(U_n)$. Here $U_n$
is the uniform distribution on $\{0,1\}^n$.
\end{definition}
A reduction from a DNP language $A$ to a DNP language $B$ using
randomized reductions $R$, $T$ is said to succeed on an input $x$ if
the
random variable $R(x)$ is distributed according to $D_2$, and for the
$y$ found serving as a witness to $R(x)$ the string $T(x, R(x), y)$
serves as a witness to $x$. If we have an $R, T$ that succeed
with probability at least $1/\hbox{poly}(n)$ for each $x$ then we have
a valid reduction from $A$ to $B$.
Now consider a universal language $\pi_{univ}$ such that
$\pi_{univ}((M, x'), y) = 1$ iff $M(x', y) = 1$ where $M$ is the
Turing-machine description of a poly-time computable function. Then
we can reduce a DNP language $(\pi, D)$ to $(\pi_{univ}, D')$ where
$D'$ has the same distribution over $x$ as $D$ and chooses the
machine $M$ according to some probability distribution such that each
machine gets picked with constant probability (e.g. the $i$th machine
is chosen with probability $2^{-i}$). We do not get a single
DNP-complete language since $D'$ differs based on the language
we are reducing from, but we do get a single language that all
DNP-problems with distribution $D$ reduce to. Impagliazzo and Levin
showed in \cite{IL90} that for every language $(\pi, D)$ there is a
language $(\tilde{\pi}, U)$ that $(\pi, D)$ reduces to, where $U$ is
the uniform distribution. By
composing the universal language reduction with their reduction, we
get that $(\pi_{univ}, U)$ is DNP-complete. The rest of this
lecture describes the proof of their result.
\section{A Special Case of [Impagliazzo, Levin]}
Recall that for a language $(\pi, D)$ there is a uniform poly-time
algorithm $G$ for sampling from $D$. If $G$ were $1$-to-$1$ then $D =
U$, so $(\pi, D)$ reduces to $(\pi_{univ}, U)$. Now we discuss the
case where $G$ is $2^{\ell}$-to-$1$ and we know $\ell$. The idea for
dealing with this case is to reduce it to the $1$-to-$1$ case.
We use pairwise independent hashing. If $G$ is $2^{\ell}$-to-$1$ then
its image size is $2^{n - \ell}$. We pick a random pairwise
independent hash function $h$ from $\{0,1\}^n$ to $\{0,1\}^{n-\ell +
2}$ (the reason for the ``$2$'' will become clear later) and
hope that we have no collisions. Our input to the new problem
$\tilde{\pi}$ will be $(h, h(x))$ where $\tilde{\pi}((h,z), (s,y)) =
1$ iff $h(G(s)) = z$ and $\pi(G(s), y) = 1$. To map the witness $(s,
y)$ back to our original problem $\pi$, we set $T(x, s, y) = y$ if $x
= G(s)$ (i.e. a witness for $\tilde{\pi}$ was not found for some
$x'\neq x$
that collided with $x$ under $h$); otherwise we label our attempt at a
reduction as a failure.
To bound the success of our reduction, we first need the following
claim.
\begin{claim}\label{goodclaim}
$\hbox{Pr}[\forall x'\in G(\{0,1\}^n) - \{x\},\ h(x') \neq h(x)] \ge 3/4$.
\end{claim}
\begin{proof}
Fix $x' \neq x$. Since $h$ is pairwise independent we have
$\hbox{Pr}[h(x') = h(x)] \le 1 / |\hbox{range}(h)|$, so by the union
bound we have $\hbox{Pr}[\exists x'\in G(\{0,1\}^n) - \{x\},\ h(x') =
h(x)] \le |G(\{0,1\}^n)| / |\hbox{range}(h)| = 2^{n - \ell} /
2^{n-\ell + 2} = 1/4$.
\end{proof}
Now to analyze the probability of success of our reduction, let $E$ be
the event that $(h,h(x))$ uniquely specifies $x$. By Claim
\ref{goodclaim} $\hbox{Pr}[E] \ge 3/4$. If $\tilde{\pi}$ is easy,
then there is some AvgBPP algorithm $A$ that fails to find a witness
on a negligible fraction of $(h,z)$ pairs (where $(h,z)$ is drawn
under the uniform distribution). Let $B$ be this set of $(h,z)$
pairs where $A$ fails so that $\hbox{Pr}_{h,z}[(h,z)\in B] =
\delta$ is negligible. Then we have
\begin{eqnarray}
\hbox{Pr}_{x\in G(\{0,1\}^n), h}\left[(h, h(x)) \in B\right] &\le&
\hbox{Pr}_{x,h}\left[(h, h(x))\in B|E\right]\cdot\hbox{Pr}[E] +
\hbox{Pr}\left[\neg E\right] \\
&=& \hbox{Pr}_{h,z}[(h,z)\in B | E]\cdot\hbox{Pr}[E] + \hbox{Pr}[\neg
E] \\
&\le& \frac{\hbox{Pr}_{h,z}[(h,z)\in
B]}{\hbox{Pr}[E]}\cdot\hbox{Pr}[E] + \hbox{Pr}[\neg E]\\
&\le& \delta + 1/4
\end{eqnarray}
Line (2) follows from (1) since the event $E$ occurring implies $z$
uniquely specifies $x$.
Now the probability that our reductions fails to work is at most\\
$\hbox{Pr}_h[h(x)\hbox{ does not uniquely specify } x] +
\hbox{Pr}_{x,h}[(h, h(x)) \in B]$ by the union bound. By Claim
\ref{goodclaim} and the above analysis, this quantity is at most $1/4
+ (\delta + 1/4) = 1/2 + \delta$, which is at least
$1/\hbox{poly}(n)$, and thus $(\pi, D)$ reduces to $(\tilde{\pi}, U)$.
\section{The General Case}
In the previous section we assumed that $G$ was $2^{\ell}$-to-$1$ and
that we knew $\ell$. In reality $G$ can be any function from
$\{0,1\}^n$ to $\{0,1\}^n$, and there might not even be an ``$\ell$'' to
know! The idea of \cite{IL90} to overcome this obstacle might be
reminiscent of the protocol of Goldwasser
and Sipser \cite{GS86} for approximating the size of a set.
We do the following upon being given an input $x$ for $(\pi, D)$:
\begin{enumerate}
\item Guess $\ell \in [0, n]$ at random. In the analysis, you should
think about ``the right $\ell$'' to guess being the $\ell$ such that
there are approximately $2^{n - \ell}$ other $y$'s such that
$|G^{-1}(x)| \approx |G^{-1}(y)|$ (within a factor of $2$).
\item Guess $k \in [0, n]$ at random such that $|\{s | G(s) = x\}| \approx 2^k$
(again,
within a factor of $2$).
\item Pick a random pairwise independent hash function $h : \{0,1\}^n
\rightarrow
\{0,1\}^{n-\ell + O(1)}$.
\item Pick a random pairwise independent hash function $\tilde{h}:
\{0,1\}^n\rightarrow \{0,1\}^{k + O(1)}$.
\item Pick $w\in\{0,1\}^k$ uniformly at random.
\end{enumerate}
Now the input of our reduction to $\tilde{\pi}$ is
$(k,\ell,h,h(x),\tilde{h},w)$. Our new language $(\tilde{\pi}, U)$
will be such that $\tilde{\pi}((h,z,\tilde{h},w), (s,y)) = 1$ iff $h(G(s))
= z$, $\pi(G(s), y) = 1$, and $\tilde{h}(s) = w$. We then transform a
witness $(s,y)$ for $\tilde{\pi}$ to a witness for $\pi$ by outputting
$y$ if $G(s) = x$ and labeling our reduction attempt a failure
otherwise.
The analysis of the general case conditions on $h, z, \tilde{h},$ and
$w$ uniquely specifying $s$ then proceeds as in
Section 3. We omit the
details, but it can be shown that $s$ being specified uniquely happens
with
non-negligible probability (with probability at least
$\Omega(1/n^2)$ --- essentially guessing $k,\ell$ is the limiting factor).
This completes the proof that $(\pi_{univ}, U)$ is DNP-complete. We
glossed over one detail and that is how to represent the $(M,x)$, the
inputs to $\pi_{univ}$, as single
strings. This can be done by writing $x' = (\langle M \rangle, x)$
where $\langle M \rangle$ is a prefix-free encoding of the description
of the machine $M$. A prefix-free encoding is a mapping from integers to
$\{0,1\}^*$ such that no encoding of one integer is a prefix of the
encoding of another integer. A simple such encoding is to map the
integer represented in binary as $b_1b_2\ldots b_k$ to the string
$b_1b_1b_2b_2\ldots b_{k-1}b_{k-1}b_k\overline{b_k}$.
\begin{thebibliography}{1}
\bibitem{IL90}
Russell~Impagliazzo, Leonid~A.~Levin. No Better Ways to Generate Hard
NP Instances than Picking Uniformly at Random. In {\em Proc. 31st
Annual Symp. on Foundations of Computer Science (FOCS)}, pages
812--821, 1990.
\bibitem{GS86}
Shafi~Goldwasser, Michael~Sipser. Private Coins versus Public Coins in
Interactive Proof Systems. In {\em Proc. 18th Annual ACM Symp. on
Theory of Computing (STOC)}, pages 59--68, 1986.
\end{thebibliography}
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