188 lines
4.9 KiB
TeX
Executable File
188 lines
4.9 KiB
TeX
Executable File
|
|
\section{Applications}
|
|
|
|
|
|
\problem{Maximum Cardinality Matching}
|
|
|
|
A \textit{matching} is a subset of edges in a bipartite graph. Nodes in a matching must not have more than one edge connected to them. \\
|
|
A matching is \textit{maximal} if it has more edges than any other matching.
|
|
|
|
\vspace{5mm}
|
|
|
|
\begin{minipage}[t]{0.48\textwidth}
|
|
\begin{center}
|
|
Initial Graph \\
|
|
\vspace{2mm}
|
|
\begin{tikzpicture}
|
|
% Nodes
|
|
\begin{scope}[layer = nodes]
|
|
\node[main] (A1) at (0mm, 24mm) {};
|
|
\node[main] (A2) at (0mm, 18mm) {};
|
|
\node[main] (A3) at (0mm, 12mm) {};
|
|
\node[main] (A4) at (0mm, 6mm) {};
|
|
\node[main] (A5) at (0mm, 0mm) {};
|
|
\node[main] (B1) at (20mm, 24mm) {};
|
|
\node[main] (B2) at (20mm, 18mm) {};
|
|
\node[main] (B3) at (20mm, 12mm) {};
|
|
\node[main] (B4) at (20mm, 6mm) {};
|
|
\node[main] (B5) at (20mm, 0mm) {};
|
|
\end{scope}
|
|
|
|
% Edges
|
|
\draw
|
|
(A1) edge (B2)
|
|
(A1) edge (B3)
|
|
(A2) edge (B1)
|
|
(A2) edge (B4)
|
|
(A4) edge (B3)
|
|
(A2) edge (B3)
|
|
(A5) edge (B3)
|
|
(A5) edge (B4)
|
|
;
|
|
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\end{minipage}
|
|
\hfill
|
|
\begin{minipage}[t]{0.48\textwidth}
|
|
\begin{center}
|
|
Maximal Matching \\
|
|
\vspace{2mm}
|
|
\begin{tikzpicture}
|
|
% Nodes
|
|
\begin{scope}[layer = nodes]
|
|
\node[main] (A1) at (0mm, 24mm) {};
|
|
\node[main] (A2) at (0mm, 18mm) {};
|
|
\node[main] (A3) at (0mm, 12mm) {};
|
|
\node[main] (A4) at (0mm, 6mm) {};
|
|
\node[main] (A5) at (0mm, 0mm) {};
|
|
\node[main] (B1) at (20mm, 24mm) {};
|
|
\node[main] (B2) at (20mm, 18mm) {};
|
|
\node[main] (B3) at (20mm, 12mm) {};
|
|
\node[main] (B4) at (20mm, 6mm) {};
|
|
\node[main] (B5) at (20mm, 0mm) {};
|
|
\end{scope}
|
|
|
|
% Edges
|
|
\draw[opacity = 0.4]
|
|
(A1) edge (B2)
|
|
(A1) edge (B3)
|
|
(A2) edge (B1)
|
|
(A2) edge (B4)
|
|
(A4) edge (B3)
|
|
(A4) edge (B3)
|
|
(A5) edge (B3)
|
|
(A5) edge (B4)
|
|
;
|
|
\draw
|
|
(A1) edge (B2)
|
|
(A2) edge (B1)
|
|
(A4) edge (B3)
|
|
(A5) edge (B4)
|
|
;
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\end{minipage}
|
|
|
|
\vspace{5mm}
|
|
|
|
Devise an algorithm to find a maximal matching in any bipartite graph. \\
|
|
Find an upper bound for its runtime.
|
|
|
|
\begin{solution}
|
|
Turn this into a maximum flow problem and use FF. \\
|
|
Connect a node $S$ to all nodes in the left group and a node $T$ to all nodes in the right group. All edges have capacity 1.
|
|
|
|
\vspace{2ex}
|
|
|
|
Just like FF, this algorithm will take at most $\min(\# \text{ left nodes}, \# \text{ right nodes})$ iterations.
|
|
\end{solution}
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\problem{Circulations with Demand}
|
|
|
|
Say we have a network of cities and power stations. Stations produce power; cities consume it.
|
|
|
|
Each station produces a limited amount of power, and each city has limited demand.
|
|
|
|
\vspace{2ex}
|
|
|
|
We can represent this power grid as a graph, with cities and stations as nodes and transmission lines as edges.
|
|
|
|
\vspace{2ex}
|
|
|
|
A simple example is below. There are two cities ($2$ and $4$) and two stations (both $-3$).
|
|
|
|
We'll represent station capacity with a negative number, since they \textit{consume} a negative amount of energy.
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}[
|
|
node distance = 25mm,
|
|
main/.style = {
|
|
draw,
|
|
circle,
|
|
fill = white,
|
|
minimum size = 8mm
|
|
},
|
|
]
|
|
% Nodes
|
|
\begin{scope}[layer = nodes]
|
|
\node[main] (S1) {$-3$};
|
|
\node[main] (S2) [below left of = S1] {$-3$};
|
|
\node[main] (C1) [below right of = S1] {$2$};
|
|
\node[main] (C2) [below right of = S2] {$4$};
|
|
\end{scope}
|
|
|
|
% Edges
|
|
\draw[->]
|
|
(S1) edge node[label] {$3$} (S2)
|
|
(S1) edge node[label] {$3$} (C1)
|
|
(S2) edge node[label] {$2$} (C1)
|
|
(S2) edge node[label] {$2$} (C2)
|
|
(C1) edge node[label] {$2$} (C2)
|
|
;
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
We'd like to know if there exists a \textit{feasible circulation} in this network---that is, can we supply our cities with the energy they need without exceeding the capacity of power plants or transmission lines?
|
|
|
|
\vspace{2ex}
|
|
|
|
\textbf{Your job:} Devise an algorithm that solve this problem.
|
|
|
|
\vspace{2ex}
|
|
|
|
\note{\textbf{Bonus:} Say certain edges have a lower bound on their capacity, meaning that we must send \textit{at least} that much flow down the edge. Modify your algorithm to account for these additional constraints.}
|
|
|
|
|
|
\begin{solution}
|
|
Create a source node $S$, and connect it to each station with an edge. Set the capacity of that edge to the capacity of the station. \\
|
|
|
|
Create a sink node $T$ and do the same for cities.
|
|
|
|
\vspace{2ex}
|
|
|
|
This is now a maximum-flow problem with one source and one sink. Apply FF.
|
|
|
|
\linehack{}
|
|
|
|
To solve the bonus problem, we'll modify the network before running the algorithm above.
|
|
|
|
\vspace{2ex}
|
|
|
|
Say an edge from $A$ to $B$ has minimum capacity $l$ and maximum capacity $u \geq l$. Apply the following transformations:
|
|
\begin{itemize}
|
|
\item Add $l$ to the capacity of $A$
|
|
\item Subtract $l$ from the capacity of $B$
|
|
\item Subtract $l$ from the total capacity of the edge.
|
|
\end{itemize}
|
|
|
|
Do this for every edge that has a lower bound then apply the algorithm above.
|
|
|
|
\end{solution}
|
|
|
|
\vfill
|
|
\pagebreak
|