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   \begin{center}
   \framebox{
      \vbox{\vspace{2mm}
    \hbox to 6.28in { {\bf CS~153~~~Current topics in theoretical
        computer science
                        \hfill Spring 2018} }
       \vspace{4mm}
       \hbox to 6.28in { {\Large \hfill Problem Set #1  \hfill} }
       \vspace{2mm}
       \hbox to 6.28in { {\it Out: #2 \hfill Due: #3} }
      \vspace{2mm}}
   }
   \end{center}
   \vspace*{4mm}
}

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% **** IF YOU WANT TO DEFINE ADDITIONAL MACROS FOR YOURSELF, PUT THEM HERE:


\begin{document}
\problemset{1}{May 1}{\bf {May 11}}

You are encouraged to work in groups of two or three;
however you must turn in your own write-up and note with whom you
worked. Please don't consult online solutions or original research
papers or surveys containing such solutions while doing this problem set. Please attempt all problems. 


\begin{enumerate}
\item Show that if $f:X \times Y \rightarrow \{0,1\}$ has a fooling
  set of size $t$, then $R_0^{\mbox{pub}}(f) \ge \log t$. Give an
  example of a function exhibiting an exponential gap between
  $R_0^{\mbox{pub}}$ and $R_{1/3}^{\mbox{pub}}$. 

\item This problem concerns the function $\mbox{TRIBES}:\{0,1\}^n \times
  \{0,1\}^n \rightarrow \{0,1\}$. We view Alice's input $x$ as a
  $\sqrt{n} \times \sqrt{n}$ matrix and Bob's input $y$ in the same
  way. Then \[\mbox{TRIBES}(x,y) = \bigwedge_i \bigvee_j (x_{i,j}\land
  y_{i,j}).\]

\begin{enumerate}
\item Prove that $D(\mbox{TRIBES}) = \Theta(n)$. 
\item Prove that $N^1(\mbox{TRIBES}) = \Theta(\sqrt{n}\log n)$. Hint:
  for the lower bound, find a $1$-fooling set of size
  $\sqrt{n}^{\sqrt{n}}$.
\item Prove that $N^0(\mbox{TRIBES}) = \Theta(\sqrt{n})$.
\end{enumerate}

\item Recall that in the $\mbox{CIS}_G$ problem, Alice holds a clique in an
  $n$ node  graph $G$ and Bob holds an independent set in $G$. Recall that we
  proved an upper bound $D(\mbox{CIS}_G) \le O(\log^2 n)$. 

\begin{enumerate}
\item Prove the lower bound $D(\mbox{CIS}_G) \ge \Omega(\log n)$, for some
  $G$.

\item Suppose that we discover a better upper bound $D(\mbox{CIS}_G) \le O(\log^c
  n)$ (that holds for all $G$). Prove that in this scenario $D(f) =
  O(\log^c C^D(f))$, where $C^D(f)$ is the size of the smallest disjoint
  cover of $f$ by monochromatic rectangles.
\end{enumerate}

\end{enumerate}

\end{document}
