% Chris Hanusa
% chanusa@qc.cuny.edu	
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\begin{center}\large 
MATH 634, Spring 2014

{\sc Homework 15}

due 5:00{\sc pm} on Monday, May 5.
\end{center}


%\noindent
%{\em Background reading:} {\em Pearls in Graph Theory}, Section 7.2.

\begin{enumerate}
\item[\bf 15D$^*$.]  Work to understand the following definitions.  \newline It is not required to turn in written descriptions.

\vspace{-.1in}
\begin{itemize}
\item MIN CUT (aka minimum $st$-cut)
\item flow companion graph
\item flow augmenting path
\end{itemize}




\item[\bf 15GS.] Run the Gale-Shapley algorithm twice on the following sets of preferences, once with the people proposing, and once with the pets proposing. Discuss how your results are related to the theorems on optimality and pessimality.
\vspace{-.1in}
\begin{center}
\begin{tabular}{c}
Pets' Preferences \\
\begin{tabular}{r|ccccc}
& Art & Bettie & Casper & Delphi & Ernie \\ \hline
1st Choice	&F	&F	&G	&G	&H \\
2nd Choice	&G	&H	&I	&H	&F \\
3rd Choice	&I	&I	&F	&J	&I \\
4th Choice	&J	&G	&H	&I	&G \\
5th Choice	&H	&J	&J	&F	&J \\
\end{tabular}
\end{tabular}
 
\begin{tabular}{c}
People's Preferences \\
\begin{tabular}{r|ccccc}
 & Fei	& Georg	& Helen	& Ivan	& Jin \\ \hline
1st Choice	&C	&E	&A	&E	&B\\
2nd Choice	&B	&B	&D	&C	&D\\
3rd Choice	&E	&C	&B	&A	&A\\
4th Choice	&A	&A	&E	&B	&C\\
5th Choice	&D	&D	&C	&D	&E\\
\end{tabular}
\end{tabular}
\end{center}

\item[\bf 15ST.]
\begin{enumerate}
\item List all $st$-cuts in the network pictured below (there are eight). Find the capacity of each $st$-cut and determine the min cut from this information. 
\item Find a maximum flow for the network (and verify that it is a max flow).  Then verify that the max flow / min cut theorem holds. 

 [{\em {\bf Important:} Do NOT use the Ford-Fulkerson algorithm in this problem.}] 
\end{enumerate}

\vspace{-.3in}
\begin{center}
\includegraphics[height=2in]{figures/hwNetworkCut.eps}
\end{center}

\vspace{-.3in}
\item[\bf 15FF.]
  Use the Ford-Fulkerson algorithm to find the max flow and min cut on the attached network.  You must prove that your flow is a max flow and your $st$-cut is a min cut.


\end{enumerate}


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