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   \noindent
   \begin{center}
   \framebox{
      \vbox{\vspace{2mm}
    \hbox to 6.28in { {\bf CS 262: Computational Complexity
                        \hfill Due: March 8, 2010} }
       \vspace{4mm}
       \hbox to 6.28in { \hfill {\Large Homework 6} \hfill
               }
       \vspace{2mm}
       \hbox to 6.28in { {\it Instructor: Sandy Irani} \hfill }
      \vspace{2mm}}
   }
   \end{center}
   \vspace*{4mm}

Do three of the following four  problems:

\begin{enumerate}

\item
Prove that if ${\bf NP} \subseteq {\bf BPP}$, then
${\bf NP} = {\bf RP}$. ({\em Hint:} use your {\bf BPP}
algorithm for SAT to give an {\bf RP} algorithm for SAT
by constructing a satisfying assignment).

\item
Show that {\bf BPP} and {\bf RP} are closed under reductions.

\item Consider the problem whose input is a graph with integer weights and
asks whether the minimum length tour is unique.
For what class in the polynomial hierarchy is this problem complete?
Prove your answer.

\item
Show that if ${\bf NP} \subseteq {\bf TIME}(n^{\log n})$, 
then ${\bf PH} \subseteq \cup_{k \ge 1}{\bf TIME}(n^{\log^k n})$.



\end{enumerate}

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