handouts/Advanced/Graph Algorithms/parts/06 bonus.tex

\section{Crosses}

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.

\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

\problem{}
Turn this into a network flow problem that can be solved with the Ford-Fulkerson algorithm.

\vfill
\pagebreak