\newcounter{lecture} \setcounter{lecture}{7}
\def\thetoday{January 29, 2009}
\def\thetitle{More Network Flow Applications}

\input{lec_hdr.tex}

Chapter $7$ in Kleiberg and Tardos [KT] discusses many applications of 
network flow to solve different types of problems. These include Project 
Scheduling, Baseball Elimination and Airline Flight Scheduling covered in 
this lecture.

\subsubsection*{Scheduling  -- Project Selection}


{\bf Problem:} We are given:
\begin{itemize}
\item a set of projects $T_1, \ldots, T_n$,
\item project $T_i$ has a profit $P_i$ which can be positive or
  negative,
\item a project can be a prerequisite of another.
\end{itemize}

{\bf Goal:}  choose a subset of projects that satisfy the prerequisite
requirement and have the maximum total profit.

{\bf Solution:} For every project $j$ with positive profit, put an
edge from $T_j$ to the target node $t$ with capacity $P_j$.  For each
project $i$ with negative profit, put an edge from the source node $s$
to $T_i$.  For each prerequiste, put an edge with capacity infinity.
Find the minimum s-t cut $(A,B)$ in this network.

\begin{claim} Jobs in $B$ have all their prerequisites met.
\end{claim}
\begin{proof}
Suppose not.  Then there is an edge with capacity $\infty$ going from
$A$ to $B$.  This cannot be, since $(A,B)$ is a min-cut.
\end{proof}

\begin{claim} $B$ is an optimal job-set.
\end{claim}
\begin{proof}
  Let $I$ be the total possible surpluses, i.e. the total weight of
  edges going into $t$.  Then total amount of profit for
  job-set $B$ is equal to $I - c(A,B)$, because edges in the cut
  either represent costs or potential surplus jobs.  Therefore,
  minimizing the cut maximizes profits.
\end{proof}

\subsubsection*{Baseball Elimination}

{\bf Problem:} Suppose we have a set of teams $T_1, \ldots, T_j$ each with a win
record $W_1, \ldots, T_j$, and a schedule of $k$ games remaining in the
season $G_1, \ldots, G_k$, where each $G_i$ is an unordered pair of
teams.  The question is, does there exist a scenario under which team
$T_j$ ends the season with at least as many wins as any other team?  In
other words, has $T_j$ been eliminated from first place?

{\bf Solution:} We will represent this as a flow problem.  First, we
can assume that $T_j$ wins its remaining $p$ games, so credit $T_j$ with
these wins and remove those games from the schedule.  Next, for each
team $i \neq j$ put a vertex and attach a directed edge to the sink
with capacity equal to the maximum number of games it can win and not
exceed $T_j$'s maximum win total.  For each game $G_i$, put an edge
from the source with capacity $1$, and put an edge to each
participating team with capacity $1$.

\begin{claim} 
$T_j$ has not been eliminated if and only if the preceding graph has a
flow with value $k-p$.
\end{claim}
\begin{proof}
  ``$\Leftarrow$'' Since each game has capacity $1$, the
  integral flow theorem a flow  implies that a flow of value $k-p$
  assigns each game to a team without exceeding their maximum win
  totals.

``$\Rightarrow$'' If $T_j$ has not been eliminated, there exists an
assignment of games to teams that does not exceed their maximum win
totals.  This defines a flow of value $k-p$.

\end{proof}

\subsubsection*{Scheduling  --  Airline Flights}

This problem modifies the original network flow problem by introducing lower
bounds on flow in addition to capacities on the edges. To see a treatment of
this and other extensions as well as how to solve this problem see Section 
$7.7$ in Kleinberg and Tardos [KT].







\end{document}