Combinatorics (Fall 2010)/Flow and matching: Difference between revisions
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imported>WikiSysop Created page with '== Flow == === Maximum Flow Problem === An instance of the maximum flow problem consists of: * a directed graph <math>G(V,E)</math>; * two distinguished vertices <math>s</math> …' |
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== Flow == | == Flow == | ||
=== | === The maximum flow problem === | ||
An instance of the maximum flow problem consists of: | An instance of the maximum flow problem consists of: | ||
* a directed graph <math>G(V,E)</math>; | * a directed graph <math>G(V,E)</math>; | ||
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The '''value''' of the flow <math>f</math> is <math>\sum_{v:(s,v)\in E}f_{sv}</math>. Given a flow network, the maximum flow problem asks to find the flow of the maximum value. | The '''value''' of the flow <math>f</math> is <math>\sum_{v:(s,v)\in E}f_{sv}</math>. Given a flow network, the maximum flow problem asks to find the flow of the maximum value. | ||
=== Cuts === | |||
=== The augmenting paths === | |||
=== The max-flow min-cut theorem === | |||
== Unimodularity == | |||
=== Integrality of polytopes === | |||
=== Unimodularity and total unimodularity === | |||
Revision as of 09:31, 18 December 2010
Flow
The maximum flow problem
An instance of the maximum flow problem consists of:
- a directed graph [math]\displaystyle{ G(V,E) }[/math];
- two distinguished vertices [math]\displaystyle{ s }[/math] (the source) and [math]\displaystyle{ t }[/math] (the sink), where the in-degree of [math]\displaystyle{ s }[/math] and the out-degree of [math]\displaystyle{ t }[/math] are both 0;
- the capacity function [math]\displaystyle{ c:E\rightarrow\mathbb{R}^+ }[/math] which associates each directed edge [math]\displaystyle{ (u,v)\in E }[/math] a nonnegative real number [math]\displaystyle{ c_{uv} }[/math] called the capacity of the edge.
The quadruple [math]\displaystyle{ (G,c,s,t) }[/math] is called a flow network.
A function [math]\displaystyle{ f:E\rightarrow\mathbb{R}^+ }[/math] is called a flow (or an [math]\displaystyle{ s }[/math]-[math]\displaystyle{ t }[/math] flow) in the network [math]\displaystyle{ G(V,E) }[/math] if it satisfies:
- Capacity constraint: [math]\displaystyle{ f_{uv}\le c_{uv} }[/math] for all [math]\displaystyle{ (u,v)\in E }[/math].
- Conservation constraint: [math]\displaystyle{ \sum_{u:(u,v)\in E}f_{uv}=\sum_{w:(v,w)\in E}f_{vw} }[/math] for all [math]\displaystyle{ v\in V\setminus\{s,t\} }[/math].
The value of the flow [math]\displaystyle{ f }[/math] is [math]\displaystyle{ \sum_{v:(s,v)\in E}f_{sv} }[/math]. Given a flow network, the maximum flow problem asks to find the flow of the maximum value.