Skip to main content Contents Index Prev Up Next Scratch ActiveCode Profile \(\newcommand{\set}[1]{\{1,2,\dotsc,#1\,\}}
\newcommand{\ints}{\mathbb{Z}}
\newcommand{\posints}{\mathbb{N}}
\newcommand{\rats}{\mathbb{Q}}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\complexes}{\mathbb{C}}
\newcommand{\twospace}{\mathbb{R}^2}
\newcommand{\threepace}{\mathbb{R}^3}
\newcommand{\dspace}{\mathbb{R}^d}
\newcommand{\nni}{\mathbb{N}_0}
\newcommand{\nonnegints}{\mathbb{N}_0}
\newcommand{\dom}{\operatorname{dom}}
\newcommand{\ran}{\operatorname{ran}}
\newcommand{\prob}{\operatorname{prob}}
\newcommand{\Prob}{\operatorname{Prob}}
\newcommand{\height}{\operatorname{height}}
\newcommand{\width}{\operatorname{width}}
\newcommand{\length}{\operatorname{length}}
\newcommand{\crit}{\operatorname{crit}}
\newcommand{\inc}{\operatorname{inc}}
\newcommand{\HP}{\mathbf{H_P}}
\newcommand{\HCP}{\mathbf{H^c_P}}
\newcommand{\GP}{\mathbf{G_P}}
\newcommand{\GQ}{\mathbf{G_Q}}
\newcommand{\AG}{\mathbf{A_G}}
\newcommand{\GCP}{\mathbf{G^c_P}}
\newcommand{\PXP}{\mathbf{P}=(X,P)}
\newcommand{\QYQ}{\mathbf{Q}=(Y,Q)}
\newcommand{\GVE}{\mathbf{G}=(V,E)}
\newcommand{\HWF}{\mathbf{H}=(W,F)}
\newcommand{\bfC}{\mathbf{C}}
\newcommand{\bfG}{\mathbf{G}}
\newcommand{\bfH}{\mathbf{H}}
\newcommand{\bfF}{\mathbf{F}}
\newcommand{\bfI}{\mathbf{I}}
\newcommand{\bfK}{\mathbf{K}}
\newcommand{\bfP}{\mathbf{P}}
\newcommand{\bfQ}{\mathbf{Q}}
\newcommand{\bfR}{\mathbf{R}}
\newcommand{\bfS}{\mathbf{S}}
\newcommand{\bfT}{\mathbf{T}}
\newcommand{\bfNP}{\mathbf{NP}}
\newcommand{\bftwo}{\mathbf{2}}
\newcommand{\cgA}{\mathcal{A}}
\newcommand{\cgB}{\mathcal{B}}
\newcommand{\cgC}{\mathcal{C}}
\newcommand{\cgD}{\mathcal{D}}
\newcommand{\cgE}{\mathcal{E}}
\newcommand{\cgF}{\mathcal{F}}
\newcommand{\cgG}{\mathcal{G}}
\newcommand{\cgM}{\mathcal{M}}
\newcommand{\cgN}{\mathcal{N}}
\newcommand{\cgP}{\mathcal{P}}
\newcommand{\cgR}{\mathcal{R}}
\newcommand{\cgS}{\mathcal{S}}
\newcommand{\bfn}{\mathbf{n}}
\newcommand{\bfm}{\mathbf{m}}
\newcommand{\bfk}{\mathbf{k}}
\newcommand{\bfs}{\mathbf{s}}
\newcommand{\bijection}{\xrightarrow[\text{onto}]{\text{$1$--$1$}}}
\newcommand{\injection}{\xrightarrow[]{\text{$1$--$1$}}}
\newcommand{\surjection}{\xrightarrow[\text{onto}]{}}
\newcommand{\nin}{\not\in}
\newcommand{\prufer}{\text{prüfer}}
\DeclareMathOperator{\fix}{fix}
\DeclareMathOperator{\stab}{stab}
\DeclareMathOperator{\var}{var}
\newcommand{\inv}{^{-1}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Chapter 14 Combinatorial Applications of Network Flows
Clearly finding the maximum flow in a network can have many direct applications to problems in business, engineering, and computer science. However, you may be surprised to learn that finding network flows can also provide reasonably efficient algorithms for solving combinatorial problems. In this chapter, we consider a restricted version of network flows in which each edge has capacity \(1\text{.}\) Our goal is to establish algorithms for two combinatorial problems: finding maximum matchings in bipartite graphs and finding the width of a poset as well as a minimal chain partition.
