Finished last two sections

Advanced/Graph Algorithms/main.tex

@@ -23,9 +23,7 @@
 	\input{parts/02 residual}
 	\input{parts/03 fulkerson}
 	\input{parts/04 applications}
-
+	\input{parts/05 reductions}
-
+	\input{parts/06 bonus}
-
-
 
 \end{document}

Advanced/Graph Algorithms/parts/05 reductions.tex

@@ -0,0 +1,131 @@
+\section{Reductions}
+
+\definition{Independent Sets}
+An \textit{independent set} is a set of vertices\footnotemark{} in which no two are connected. $\{B, C, D, E\}$ form an independent set in the following graph:
+
+\footnotetext{\say{Node} and \say{Vertex} are synonyms in graph theory.}
+
+
+\begin{center}
+	\begin{tikzpicture}[
+		node distance = 12mm,
+		hatch/.style = {
+			pattern=north west lines,
+			pattern color=gray
+		}
+	]
+		% Nodes
+		\begin{scope}[layer = nodes]
+			\node[main] (A) {$A$};
+
+
+			% Patterns are transparent.
+			% Fill nodes first so paths don't show through
+			\node[main, draw = white] (B1) [above left of = A] {$\phantom{B}$};
+			\node[main, draw = white] (C1) [below left of = A] {$\phantom{C}$};
+			\node[main, draw = white] (D1) [below right of = A] {$\phantom{D}$};
+			\node[main, draw = white] (E1) [above right of = A] {$\phantom{E}$};
+
+			\node[main, hatch] (B) [above left of = A] {$B$};
+			\node[main, hatch] (C) [below left of = A] {$C$};
+			\node[main, hatch] (D) [below right of = A] {$D$};
+			\node[main, hatch] (E) [above right of = A] {$E$};
+		\end{scope}
+
+		% Edges
+		\draw
+			(A) edge (B)
+			(A) edge (C)
+			(A) edge (D)
+			(A) edge (E)
+		;
+	\end{tikzpicture}
+\end{center}
+
+
+\definition{Vertex Covers}
+A \textit{vertex cover} is a set of vertices that includes at least one endpoint of each edge. $B$ and $D$ form a vertex cover of the following graph:
+
+\begin{center}
+	\begin{tikzpicture}[
+		node distance = 12mm,
+		hatch/.style = {
+			pattern=north west lines,
+			pattern color=gray
+		}
+	]
+		% Nodes
+		\begin{scope}[layer = nodes]
+			\node[main] (A) {$A$};
+
+			% Patterns are transparent.
+			% Fill nodes first so paths don't show through
+			\node[main, draw = white] (B1) [right of = A] {$\phantom{B}$};
+			\node[main, hatch] (B) [right of = A] {$B$};
+			\node[main, draw = white] (D1) [below of = B] {$\phantom{D}$};
+			\node[main, hatch] (D) [below of = B] {$D$};
+
+			\node[main] (C) [right of = B] {$C$};
+			\node[main] (E) [right of = D] {$E$};
+		\end{scope}
+
+		% Edges
+		\draw
+			(A) edge (B)
+			(B) edge (C)
+			(B) edge (D)
+			(D) edge (E)
+		;
+
+		% Flow
+		\draw[path]
+			(B) -- (A)
+			(B) -- (C)
+			(B) -- (D)
+			(D) -- (E)
+		;
+	\end{tikzpicture}
+\end{center}
+
+\vfill
+\pagebreak
+
+\problem{}<IndepCover>
+Let $G$ be a graph with a set of vertices $V$. \\
+
+Show that $S \subset V$ is an independent set iff $(V - S)$ is a vertex cover. \\
+
+\hint{$(V - S)$ is the set of elements in $V$ that are not in $S$.}
+
+\begin{solution}
+	Suppose $S$ is an independent set.
+	\begin{itemize}
+		\item [$\implies$] All edges are in $(V - S)$ or connect $(V - S)$ and $S$.
+		\item [$\implies$] $(V - S)$ is a vertex cover.
+	\end{itemize}
+
+	\linehack{}
+
+	Suppose $S$ is a vertex cover.
+	\begin{itemize}
+		\item [$\implies$] There are no edges with both endpoints in $(V - S)$.
+		\item [$\implies$] $(V - S)$ is an independent set.
+	\end{itemize}
+\end{solution}
+
+\vfill
+
+\problem{}
+Consider the following two problems:
+\begin{itemize}
+	\item Given a graph $G$, determine if it has an independent set of size $\geq k$.
+	\item Given a graph $G$, determine if it has a vertex cover of size $\leq k$.
+\end{itemize}
+Show that these are equivalent. In other words, show that an algorithm that solves one can be used to solve the other.
+
+\begin{solution}
+	This is a direct consequence of \ref{IndepCover}. You'll need to show that the size constraints are satisfied, but that's fairly easy to do.
+\end{solution}
+
+\vfill
+\pagebreak

Advanced/Graph Algorithms/parts/06 bonus.tex

@@ -0,0 +1,200 @@
+\section{Crosses (Bonus Problem)}
+
+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.
+
+\vspace{2ex}
+
+Here is an example of such a grid, including a possible solution.
+
+\newcommand{\bx}[2]{
+	\draw[
+		line width = 1.5mm
+	]
+		(#1 + 0.3, #2 + 0.3) -- (#1 + 0.7, #2 + 0.7)
+		(#1 + 0.7, #2 + 0.3) -- (#1 + 0.3, #2 + 0.7);
+}
+
+\newcommand{\dk}[2]{
+	\draw[
+		line width = 0mm,
+		fill = gray
+	]
+		(#1, #2) --
+		(#1 + 1, #2) --
+		(#1 + 1, #2 + 1) --
+		(#1, #2 + 1);
+}
+
+\begin{center}
+\begin{tikzpicture}[
+	scale = 0.8
+]
+
+		% Dark squares
+		\dk{0}{2}
+		\dk{1}{0}
+		\dk{1}{1}
+		\dk{1}{2}
+		\dk{1}{4}
+		\dk{2}{2}
+		\dk{2}{4}
+		\dk{3}{0}
+		\dk{3}{1}
+		\dk{3}{3}
+		\dk{3}{4}
+		\dk{4}{3}
+		\dk{4}{1}
+
+
+		% Base grid
+		\foreach \x in {0,...,5} {
+			\draw[line width = 0.4mm]
+				(0, \x) -- (5, \x)
+				(\x, 0) -- (\x, 5);
+		}
+
+		% X marks
+		\bx{0}{4}
+		\bx{1}{3}
+		\bx{2}{1}
+		\bx{3}{2}
+		\bx{4}{0}
+\end{tikzpicture}
+\end{center}
+
+\problem{}
+Find a solution for the following grid.
+
+\begin{center}
+\begin{tikzpicture}[
+	scale = 1
+]
+	% Dark squares
+	\dk{0}{2}
+	\dk{0}{3}
+	\dk{0}{6}
+	\dk{0}{7}
+	\dk{1}{0}
+	\dk{1}{1}
+	\dk{1}{4}
+	\dk{1}{5}
+	\dk{1}{6}
+	\dk{1}{7}
+	\dk{2}{0}
+	\dk{2}{1}
+	\dk{2}{3}
+	\dk{2}{4}
+	\dk{2}{5}
+	\dk{2}{6}
+	\dk{2}{7}
+	\dk{3}{1}
+	\dk{3}{2}
+	\dk{3}{3}
+	\dk{3}{4}
+	\dk{3}{5}
+	\dk{3}{6}
+	\dk{4}{0}
+	\dk{4}{1}
+	\dk{4}{2}
+	\dk{4}{3}
+	\dk{4}{6}
+	\dk{5}{1}
+	\dk{5}{4}
+	\dk{5}{5}
+	\dk{5}{6}
+	\dk{6}{0}
+	\dk{6}{1}
+	\dk{6}{2}
+	\dk{6}{3}
+	\dk{6}{4}
+	\dk{6}{5}
+	\dk{7}{0}
+	\dk{7}{4}
+	\dk{7}{6}
+	\dk{7}{7}
+
+	% Base grid
+	\foreach \x in {0,...,8} {
+		\draw[line width = 0.4mm]
+			(0, \x) -- (8, \x)
+			(\x, 0) -- (\x, 8);
+	}
+\end{tikzpicture}
+\end{center}
+
+\pagebreak
+
+\begin{solution}
+	\begin{center}
+	\begin{tikzpicture}[
+		scale = 0.6
+	]
+		% Dark squares
+		\dk{0}{2}
+		\dk{0}{3}
+		\dk{0}{6}
+		\dk{0}{7}
+		\dk{1}{0}
+		\dk{1}{1}
+		\dk{1}{4}
+		\dk{1}{5}
+		\dk{1}{6}
+		\dk{1}{7}
+		\dk{2}{0}
+		\dk{2}{1}
+		\dk{2}{3}
+		\dk{2}{4}
+		\dk{2}{5}
+		\dk{2}{6}
+		\dk{2}{7}
+		\dk{3}{1}
+		\dk{3}{2}
+		\dk{3}{3}
+		\dk{3}{4}
+		\dk{3}{5}
+		\dk{3}{6}
+		\dk{4}{0}
+		\dk{4}{1}
+		\dk{4}{2}
+		\dk{4}{3}
+		\dk{4}{6}
+		\dk{5}{1}
+		\dk{5}{4}
+		\dk{5}{5}
+		\dk{5}{6}
+		\dk{6}{0}
+		\dk{6}{1}
+		\dk{6}{2}
+		\dk{6}{3}
+		\dk{6}{4}
+		\dk{6}{5}
+		\dk{7}{0}
+		\dk{7}{4}
+		\dk{7}{6}
+		\dk{7}{7}
+
+		% Base grid
+		\foreach \x in {0,...,8} {
+			\draw[line width = 0.4mm]
+				(0, \x) -- (8, \x)
+				(\x, 0) -- (\x, 8);
+		}
+
+		% X marks
+		\bx{0}{5}
+		\bx{1}{3}
+		\bx{2}{2}
+		\bx{3}{7}
+		\bx{4}{4}
+		\bx{5}{0}
+		\bx{6}{6}
+		\bx{7}{1}
+	\end{tikzpicture}
+	\end{center}
+\end{solution}
+
+\problem{}
+Turn this into a network flow problem.
+
+\vfill
+\pagebreak

Advanced/Graph Algorithms/tikxset.tex

@@ -1,5 +1,6 @@
 \usetikzlibrary{arrows.meta}
 \usetikzlibrary{shapes.geometric}
+\usetikzlibrary{patterns}
 
 % We put nodes in a separate layer, so we can
 % slightly overlap with paths for a perfect fit