Applied edits to network flow handout
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\definition{}
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A \textit{graph} consists of a set of \textit{nodes} $\{A, B, ...\}$ and a set of edges $\{ (A,B), (A,C), ...\}$ connecting them.
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A \textit{directed graph} is a graph where edges have direction. In such a graph, $(A, B)$ and $(B, A)$ are distinct edges.
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A \textit{directed graph} is a graph where edges have direction. In such a graph, edges $(A, B)$ and $(B, A)$ are distinct.
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A \textit{weighted graph} is a graph that features weights on its edges. \\
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A weighted directed graph is shown below.
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@ -28,8 +28,7 @@ A weighted directed graph is shown below.
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\vfill
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\definition{}
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We say a graph is \textit{bipartite} if its nodes can be split into two groups $L$ and $R$ so that no two nodes in the same group are connected by an edge. \\
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The following graph is bipartite:
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We say a graph is \textit{bipartite} if its nodes can be split into two groups $L$ and $R$, where no two nodes in the same group are connected by an edge: \\
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\begin{center}
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\begin{tikzpicture}
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@ -39,9 +38,9 @@ The following graph is bipartite:
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\node[main] (B) at (0mm, -10mm) {$B$};
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\node[main] (C) at (0mm, -20mm) {$C$};
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\node[main] (D) at (20mm, 0mm) {$D$};
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\node[main] (E) at (20mm, -10mm) {$E$};
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\node[main] (F) at (20mm, -20mm) {$F$};
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\node[main, hatch] (D) at (20mm, 0mm) {$D$};
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\node[main, hatch] (E) at (20mm, -10mm) {$E$};
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\node[main, hatch] (F) at (20mm, -20mm) {$F$};
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\end{scope}
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% Edges
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@ -55,30 +54,5 @@ The following graph is bipartite:
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\end{tikzpicture}
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\end{center}
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\vfill
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\definition{}
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We say two nodes $A$ ane $B$ are \textit{connected} if we can reach $A$ from $B$ and $B$ from $A$ by walking along (possibly directed) edges. We say a graph is connected if all its nodes are connected to each other.\\
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The bipartite graph above and the directed graph below are not connected.
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\begin{center}
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\begin{tikzpicture}[node distance = 20mm]
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% Nodes
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\begin{scope}
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\node[main] (A) {$A$};
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\node[main] (B) [below right of = A] {$B$};
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\node[main] (C) [below left of = A] {$C$};
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\end{scope}
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% Edges
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\draw[->]
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(A) edge[bend right] (B)
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(B) edge[bend right] (A)
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(B) edge (C)
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;
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\end{tikzpicture}
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\end{center}
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\vfill
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\pagebreak
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@ -1,23 +1,23 @@
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\section{Network Flow}
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\generic{Networks}
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Say have a network: a sequence of pipes, a set of cities and highways, an electrical circuit, server infrastructure, etc.
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Say have a network: a sequence of pipes, a set of cities and highways, an electrical circuit, etc.
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\vspace{1ex}
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We'll represent our network with a connected directed weighted graph. If we take a city, edges will be highways and cities will be nodes. There are a few conditions for a valid network graph:
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We can draw this network as a directed weighted graph. If we take a transporation network, for example, edges will represent highways and nodes will be cities. There are a few conditions for a valid network graph:
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\begin{itemize}
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\item The weight of each edge represents its capacity, the number of lanes in the highway.
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\item The weight of each edge represents its capacity, e.g, the number of lanes in the highway.
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\item Edge capacities are always positive integers.\hspace{-0.5ex}\footnotemark{}
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\item Node $S$ is a \textit{source}: it produces flow.
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\item Node $T$ is a \textit{sink}: it consumes flow.
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\item All other nodes \textit{conserve} flow. In other words, the sum of flow coming in must equal the sum of flow going out.
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\item All other nodes \textit{conserve} flow. The sum of flow coming in must equal the sum of flow going out.
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\end{itemize}
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\footnotetext{An edge with capacity zero is equivalent to an edge that does not exist; An edge with negative capacity is equivalent to an edge in the opposite direction.}
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Here is an example of a such a graph:
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Here is an example of such a graph:
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\begin{center}
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\begin{tikzpicture}
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@ -43,7 +43,7 @@ Here is an example of a such a graph:
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\hrule{}
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\generic{Flow}
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In our city example, cars represent \textit{flow}. Let's send one unit of cars along the topmost highway:
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In our city example, traffic represents \textit{flow}. Let's send one unit of traffic along the topmost highway:
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\vspace{2ex}
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@ -89,11 +89,9 @@ In our city example, cars represent \textit{flow}. Let's send one unit of cars a
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\vspace{1ex}
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The \textit{magnitude} of a flow\footnotemark{} is the number of \say{flow-units} that go from $S$ to $T$. \\
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The \textit{magnitude} of a flow is the number of \say{flow-units} that go from $S$ to $T$. \\
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We are interested in the \textit{maximum flow} through this network: what is the greatest amount of flow we can push from $S$ to $T$?
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\footnotetext{you could also think of \say{flow} as a directed weighted graph on top of our network.}
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We are interested in the \textit{maximum flow} through this network: what is the greatest amount of flow we can get from $S$ to $T$?
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\problem{}
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What is the magnitude of the flow above?
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@ -156,7 +154,8 @@ Find a maximal flow on the graph below. \\
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\pagebreak
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\section{Combining Flows}
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It is fairly easy to combine two flows on a graph. All we need to do is add the flows along each edge. For example, consider the following flows:
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It is fairly easy to combine two flows on a graph. All we need to do is add the flows along each edge. \\
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Consider the following flows:
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\vspace{2ex}
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@ -266,11 +265,7 @@ It is fairly easy to combine two flows on a graph. All we need to do is add the
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\vspace{1ex}
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For example, we could not add these graphs if the magnitude of flow in the right graph above was 2.
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\vspace{1ex}
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This is because the capacity of the top-right edge is 2, and $2 + 1 > 2$.
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For example, we could not add these graphs if the magnitude of flow in the right graph above was 2, since the capacity of the top-right edge is 2, and $2 + 1 > 2$.
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\end{minipage}
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\vspace{2ex}
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@ -278,7 +273,7 @@ It is fairly easy to combine two flows on a graph. All we need to do is add the
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\vspace{2ex}
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\problem{}
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Combine the following flows and ensure that the flow along all edges remains within capacity.
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Combine the flows below, ensuring that the flow along each edge remains within capacity.
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\vspace{2ex}
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@ -1,9 +1,10 @@
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\section{Residual Graphs}
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As our network gets bigger, finding a maximum flow by hand becomes much more difficult. It will be convenient to have an algorithm that finds a maximal flow in any network.
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It is hard to find a maximum flow for a large network by hand. \\
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We need to create an algorithm to accomplish this task.
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\vspace{1ex}
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The first thing we'll need to construct such an algorithm is a \textit{residual graph}.
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The first thing we'll is the notion of a \textit{residual graph}.
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\vspace{2ex}
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\hrule
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@ -202,11 +203,9 @@ If it isn't, find a maximal flow. \\
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\problem{}
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Show that...
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\begin{enumerate}
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\item A maximal flow exists in every network with integral\footnotemark{} edge weights.
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\item A maximal flow exists in every network with integral edge weights.
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\item Every edge in this flow carries an integral amount of flow
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\end{enumerate}
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\footnotetext{Integral = \say{integer} as an adjective.}
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\vfill
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\pagebreak
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@ -108,7 +108,7 @@ There is extra space on the next page.
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\pagebreak
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\problem{}
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You are given a large network. How would you quickly find an upper bound for the number of iterations the Ford-Fulkerson algorithm will need to find a maximum flow?
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You are given a large network. How can you quickly find an upper bound for the number of iterations the Ford-Fulkerson algorithm will need to find a maximum flow?
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\begin{solution}
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Each iteration adds at least one unit of flow. So, we will find a maximum flow in at most $\min(\text{flow out of } S,~\text{flow into } T)$ iterations.
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@ -86,8 +86,9 @@ A matching is \textit{maximal} if it has more edges than any other matching.
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\vspace{5mm}
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Devise an algorithm to find a maximal matching in any bipartite graph. \\
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Find an upper bound for its runtime.
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Create an algorithm that finds a maximal matching in any bipartite graph. \\
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Find an upper bound for its runtime. \\
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\hint{Can you modify an algorithm we already know?}
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\begin{solution}
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Turn this into a maximum flow problem and use FF. \\
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@ -101,7 +102,7 @@ Find an upper bound for its runtime.
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\vfill
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\pagebreak
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\problem{Circulations with Demand}
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\problem{Supply and Demand}
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Say we have a network of cities and power stations. Stations produce power; cities consume it.
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@ -146,7 +147,7 @@ We'll represent station capacity with a negative number, since they \textit{cons
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\end{tikzpicture}
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\end{center}
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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?
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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 our power plants and transmission lines?
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\vspace{2ex}
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@ -8,11 +8,7 @@ An \textit{independent set} is a set of vertices\footnotemark{} in which no two
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\begin{center}
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\begin{tikzpicture}[
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node distance = 12mm,
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hatch/.style = {
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pattern=north west lines,
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pattern color=gray
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}
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node distance = 12mm
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]
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% Nodes
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\begin{scope}[layer = nodes]
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@ -48,11 +44,7 @@ A \textit{vertex cover} is a set of vertices that includes at least one endpoint
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\begin{center}
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\begin{tikzpicture}[
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node distance = 12mm,
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hatch/.style = {
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pattern=north west lines,
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pattern color=gray
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}
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node distance = 12mm
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]
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% Nodes
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\begin{scope}[layer = nodes]
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@ -1,6 +1,6 @@
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\section{Crosses (Bonus Problem)}
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\section{Crosses}
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You are given an $n \times n$ grid. Some of its squares are white, and some are gray. Your goal is to place $n$ crosses on white cells so that each row and each column contains exactly one cross.
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You are given an $n \times n$ grid. Some of its squares are white, some are gray. Your goal is to place $n$ crosses on white cells so that each row and each column contains exactly one cross.
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\vspace{2ex}
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@ -49,5 +49,9 @@
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% completely under nodes.
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line cap = rect,
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opacity = 0.3
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},
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hatch/.style = {
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pattern=north west lines,
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pattern color=gray
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}
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}
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