180 lines
3.8 KiB
TeX
Executable File
180 lines
3.8 KiB
TeX
Executable File
% https://git.betalupi.com/Mark/latex-packages
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% use [nosolutions] flag to hide solutions.
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% use [solutions] flag to show solutions.
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% Last built with version 1.1.0
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\documentclass[
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solutions
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]{ormc_handout}
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\usepackage{subfiles}
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\usetikzlibrary{arrows.meta}
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\usetikzlibrary{shapes.geometric}
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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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\pgfdeclarelayer{nodes}
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\pgfdeclarelayer{path}
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\pgfsetlayers{main,nodes}
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% Layer settings
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\tikzset{
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% Layer hack, lets us write
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% later = * in scopes.
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layer/.style = {
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execute at begin scope={\pgfonlayer{#1}},
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execute at end scope={\endpgfonlayer}
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},
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%
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% Arrowhead tweaks
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>={Latex[ width=2mm, length=2mm ]},
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label/.style = {
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circle,
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% For automatic red background in solutions
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fill = \ORMCbgcolor,
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draw = none
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},
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%
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% Nodes
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main/.style = {
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draw,
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circle,
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fill = white
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},
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%
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% Flow annotations
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flow/.style = {
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opacity = 1,
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thin,
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inner xsep = 2.5mm,
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inner ysep = 2.5mm
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},
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%
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% Paths
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path/.style = {
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line width = 4mm,
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draw = black,
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% Lengthen paths so they're
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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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}
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\begin{document}
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\maketitle
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<Advanced 2>
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<Fall 2022>
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{Algorithms on Graphs: Flow}
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<Part 1: Flow>
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{
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Prepared by Mark on \today
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}
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\subfile{parts/00 review}
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\subfile{parts/01 flow}
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\subfile{parts/02 residual}
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\subfile{parts/03 fulkerson}
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\problem{Maximum Cardinality Matching}
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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. \\
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A matching is \textit{maximal} if it has more edges than any other matching.
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\vspace{5mm}
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\begin{minipage}[t]{0.48\textwidth}
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\begin{center}
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Initial Graph \\
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\vspace{2mm}
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\begin{tikzpicture}
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% Nodes
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\begin{scope}[layer = nodes]
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\node[main] (A1) at (0mm, 24mm) {};
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\node[main] (A2) at (0mm, 18mm) {};
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\node[main] (A3) at (0mm, 12mm) {};
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\node[main] (A4) at (0mm, 6mm) {};
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\node[main] (A5) at (0mm, 0mm) {};
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\node[main] (B1) at (20mm, 24mm) {};
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\node[main] (B2) at (20mm, 18mm) {};
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\node[main] (B3) at (20mm, 12mm) {};
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\node[main] (B4) at (20mm, 6mm) {};
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\node[main] (B5) at (20mm, 0mm) {};
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\end{scope}
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% Edges
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\draw
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(A1) edge (B2)
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(A1) edge (B3)
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(A2) edge (B1)
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(A2) edge (B4)
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(A4) edge (B3)
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(A2) edge (B3)
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(A5) edge (B3)
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(A5) edge (B4)
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;
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.48\textwidth}
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\begin{center}
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Maximal Matching \\
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\vspace{2mm}
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\begin{tikzpicture}
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% Nodes
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\begin{scope}[layer = nodes]
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\node[main] (A1) at (0mm, 24mm) {};
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\node[main] (A2) at (0mm, 18mm) {};
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\node[main] (A3) at (0mm, 12mm) {};
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\node[main] (A4) at (0mm, 6mm) {};
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\node[main] (A5) at (0mm, 0mm) {};
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\node[main] (B1) at (20mm, 24mm) {};
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\node[main] (B2) at (20mm, 18mm) {};
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\node[main] (B3) at (20mm, 12mm) {};
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\node[main] (B4) at (20mm, 6mm) {};
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\node[main] (B5) at (20mm, 0mm) {};
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\end{scope}
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% Edges
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\draw[opacity = 0.4]
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(A1) edge (B2)
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(A1) edge (B3)
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(A2) edge (B1)
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(A2) edge (B4)
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(A4) edge (B3)
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(A4) edge (B3)
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(A5) edge (B3)
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(A5) edge (B4)
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;
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\draw
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(A1) edge (B2)
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(A2) edge (B1)
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(A4) edge (B3)
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(A5) edge (B4)
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;
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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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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\begin{solution}
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Turn this into a maximum flow problem and use FF. \\
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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.
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\vspace{2ex}
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Just like FF, this algorithm will take at most $\min(\# \text{ left nodes}, \# \text{ right nodes})$ iterations.
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\end{solution}
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\end{document} |