Finished last two sections
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\input{parts/02 residual}
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\input{parts/03 fulkerson}
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\input{parts/04 applications}
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\input{parts/05 reductions}
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\input{parts/06 bonus}
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\end{document}
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131
Advanced/Graph Algorithms/parts/05 reductions.tex
Executable file
131
Advanced/Graph Algorithms/parts/05 reductions.tex
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\section{Reductions}
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\definition{Independent Sets}
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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:
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\footnotetext{\say{Node} and \say{Vertex} are synonyms in graph theory.}
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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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]
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% Nodes
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\begin{scope}[layer = nodes]
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\node[main] (A) {$A$};
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% Patterns are transparent.
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% Fill nodes first so paths don't show through
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\node[main, draw = white] (B1) [above left of = A] {$\phantom{B}$};
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\node[main, draw = white] (C1) [below left of = A] {$\phantom{C}$};
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\node[main, draw = white] (D1) [below right of = A] {$\phantom{D}$};
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\node[main, draw = white] (E1) [above right of = A] {$\phantom{E}$};
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\node[main, hatch] (B) [above left of = A] {$B$};
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\node[main, hatch] (C) [below left of = A] {$C$};
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\node[main, hatch] (D) [below right of = A] {$D$};
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\node[main, hatch] (E) [above right of = A] {$E$};
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\end{scope}
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% Edges
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\draw
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(A) edge (B)
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(A) edge (C)
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(A) edge (D)
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(A) edge (E)
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;
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\end{tikzpicture}
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\end{center}
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\definition{Vertex Covers}
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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:
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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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]
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% Nodes
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\begin{scope}[layer = nodes]
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\node[main] (A) {$A$};
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% Patterns are transparent.
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% Fill nodes first so paths don't show through
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\node[main, draw = white] (B1) [right of = A] {$\phantom{B}$};
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\node[main, hatch] (B) [right of = A] {$B$};
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\node[main, draw = white] (D1) [below of = B] {$\phantom{D}$};
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\node[main, hatch] (D) [below of = B] {$D$};
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\node[main] (C) [right of = B] {$C$};
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\node[main] (E) [right of = D] {$E$};
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\end{scope}
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% Edges
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\draw
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(A) edge (B)
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(B) edge (C)
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(B) edge (D)
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(D) edge (E)
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;
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% Flow
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\draw[path]
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(B) -- (A)
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(B) -- (C)
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(B) -- (D)
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(D) -- (E)
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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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\problem{}<IndepCover>
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Let $G$ be a graph with a set of vertices $V$. \\
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Show that $S \subset V$ is an independent set iff $(V - S)$ is a vertex cover. \\
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\hint{$(V - S)$ is the set of elements in $V$ that are not in $S$.}
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\begin{solution}
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Suppose $S$ is an independent set.
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\begin{itemize}
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\item [$\implies$] All edges are in $(V - S)$ or connect $(V - S)$ and $S$.
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\item [$\implies$] $(V - S)$ is a vertex cover.
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\end{itemize}
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\linehack{}
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Suppose $S$ is a vertex cover.
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\begin{itemize}
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\item [$\implies$] There are no edges with both endpoints in $(V - S)$.
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\item [$\implies$] $(V - S)$ is an independent set.
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\end{itemize}
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\end{solution}
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\vfill
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\problem{}
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Consider the following two problems:
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\begin{itemize}
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\item Given a graph $G$, determine if it has an independent set of size $\geq k$.
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\item Given a graph $G$, determine if it has a vertex cover of size $\leq k$.
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\end{itemize}
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Show that these are equivalent. In other words, show that an algorithm that solves one can be used to solve the other.
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\begin{solution}
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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.
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\end{solution}
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\vfill
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\pagebreak
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200
Advanced/Graph Algorithms/parts/06 bonus.tex
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200
Advanced/Graph Algorithms/parts/06 bonus.tex
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\section{Crosses (Bonus Problem)}
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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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\vspace{2ex}
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Here is an example of such a grid, including a possible solution.
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\newcommand{\bx}[2]{
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\draw[
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line width = 1.5mm
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]
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(#1 + 0.3, #2 + 0.3) -- (#1 + 0.7, #2 + 0.7)
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(#1 + 0.7, #2 + 0.3) -- (#1 + 0.3, #2 + 0.7);
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}
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\newcommand{\dk}[2]{
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\draw[
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line width = 0mm,
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fill = gray
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]
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(#1, #2) --
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(#1 + 1, #2) --
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(#1 + 1, #2 + 1) --
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(#1, #2 + 1);
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}
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\begin{center}
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\begin{tikzpicture}[
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scale = 0.8
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]
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% Dark squares
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\dk{0}{2}
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\dk{1}{0}
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\dk{1}{1}
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\dk{1}{2}
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\dk{1}{4}
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\dk{2}{2}
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\dk{2}{4}
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\dk{3}{0}
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\dk{3}{1}
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\dk{3}{3}
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\dk{3}{4}
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\dk{4}{3}
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\dk{4}{1}
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% Base grid
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\foreach \x in {0,...,5} {
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\draw[line width = 0.4mm]
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(0, \x) -- (5, \x)
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(\x, 0) -- (\x, 5);
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}
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% X marks
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\bx{0}{4}
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\bx{1}{3}
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\bx{2}{1}
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\bx{3}{2}
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\bx{4}{0}
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\end{tikzpicture}
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\end{center}
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\problem{}
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Find a solution for the following grid.
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\begin{center}
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\begin{tikzpicture}[
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scale = 1
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]
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% Dark squares
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\dk{0}{2}
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\dk{0}{3}
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\dk{0}{6}
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\dk{0}{7}
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\dk{1}{0}
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\dk{1}{1}
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\dk{1}{4}
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\dk{1}{5}
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\dk{1}{6}
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\dk{1}{7}
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\dk{2}{0}
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\dk{2}{1}
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\dk{2}{3}
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\dk{2}{4}
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\dk{2}{5}
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\dk{2}{6}
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\dk{2}{7}
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\dk{3}{1}
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\dk{3}{2}
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\dk{3}{3}
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\dk{3}{4}
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\dk{3}{5}
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\dk{3}{6}
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\dk{4}{0}
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\dk{4}{1}
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\dk{4}{2}
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\dk{4}{3}
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\dk{4}{6}
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\dk{5}{1}
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\dk{5}{4}
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\dk{5}{5}
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\dk{5}{6}
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\dk{6}{0}
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\dk{6}{1}
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\dk{6}{2}
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\dk{6}{3}
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\dk{6}{4}
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\dk{6}{5}
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\dk{7}{0}
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\dk{7}{4}
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\dk{7}{6}
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\dk{7}{7}
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% Base grid
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\foreach \x in {0,...,8} {
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\draw[line width = 0.4mm]
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(0, \x) -- (8, \x)
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(\x, 0) -- (\x, 8);
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}
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\end{tikzpicture}
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\end{center}
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\pagebreak
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\begin{solution}
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\begin{center}
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\begin{tikzpicture}[
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scale = 0.6
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]
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% Dark squares
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\dk{0}{2}
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\dk{0}{3}
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\dk{0}{6}
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\dk{0}{7}
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\dk{1}{0}
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\dk{1}{1}
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\dk{1}{4}
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\dk{1}{5}
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\dk{1}{6}
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\dk{1}{7}
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\dk{2}{0}
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\dk{2}{1}
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\dk{2}{3}
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\dk{2}{4}
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\dk{2}{5}
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\dk{2}{6}
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\dk{2}{7}
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\dk{3}{1}
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\dk{3}{2}
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\dk{3}{3}
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\dk{3}{4}
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\dk{3}{5}
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\dk{3}{6}
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\dk{4}{0}
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\dk{4}{1}
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\dk{4}{2}
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\dk{4}{3}
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\dk{4}{6}
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\dk{5}{1}
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\dk{5}{4}
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\dk{5}{5}
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\dk{5}{6}
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\dk{6}{0}
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\dk{6}{1}
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\dk{6}{2}
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\dk{6}{3}
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\dk{6}{4}
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\dk{6}{5}
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\dk{7}{0}
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\dk{7}{4}
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\dk{7}{6}
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\dk{7}{7}
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% Base grid
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\foreach \x in {0,...,8} {
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\draw[line width = 0.4mm]
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(0, \x) -- (8, \x)
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(\x, 0) -- (\x, 8);
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}
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% X marks
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\bx{0}{5}
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\bx{1}{3}
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\bx{2}{2}
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\bx{3}{7}
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\bx{4}{4}
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\bx{5}{0}
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\bx{6}{6}
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\bx{7}{1}
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\end{tikzpicture}
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\end{center}
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\end{solution}
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\problem{}
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Turn this into a network flow problem.
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\vfill
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\pagebreak
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@ -1,5 +1,6 @@
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\usetikzlibrary{arrows.meta}
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\usetikzlibrary{shapes.geometric}
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\usetikzlibrary{patterns}
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% We put nodes in a separate layer, so we can
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% slightly overlap with paths for a perfect fit
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