Maximum Flow Algorithms

Detailed Description

This group contains the algorithms for finding maximum flows and feasible circulations [6], [1].

The maximum flow problem is to find a flow of maximum value between a single source and a single target. Formally, there is a \(G=(V,A)\) digraph, a \(cap: A\rightarrow\mathbf{R}^+_0\) capacity function and \(s, t \in V\) source and target nodes. A maximum flow is an \(f: A\rightarrow\mathbf{R}^+_0\) solution of the following optimization problem.

\[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \]

\[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu) \quad \forall u\in V\setminus\{s,t\} \]

\[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \]

Preflow is an efficient implementation of Goldberg-Tarjan's preflow push-relabel algorithm [15] for finding maximum flows. It also provides functions to query the minimum cut, which is the dual problem of maximum flow.

Circulation is a preflow push-relabel algorithm implemented directly for finding feasible circulations, which is a somewhat different problem, but it is strongly related to maximum flow. For more information, see Circulation.

Classes
class	Circulation< GR, LM, UM, SM, TR >
	Push-relabel algorithm for the network circulation problem. More...
class	EdmondsKarp< GR, CAP, TR >
	Edmonds-Karp algorithms class. More...
class	Preflow< GR, CAP, TR >
	Preflow algorithm class. More...
struct	Circulation< GR, LM, UM, SM, TR >::SetElevator< T >
	Named parameter for setting Elevator type More...
struct	Circulation< GR, LM, UM, SM, TR >::SetFlowMap< T >
	Named parameter for setting FlowMap type More...
struct	Circulation< GR, LM, UM, SM, TR >::SetStandardElevator< T >
	Named parameter for setting Elevator type with automatic allocation More...
struct	EdmondsKarp< GR, CAP, TR >::SetFlowMap< T >
struct	Preflow< GR, CAP, TR >::SetElevator< T >
	Named parameter for setting Elevator type More...
struct	Preflow< GR, CAP, TR >::SetFlowMap< T >
	Named parameter for setting FlowMap type More...
struct	Preflow< GR, CAP, TR >::SetStandardElevator< T >
	Named parameter for setting Elevator type with automatic allocation More...

Files
file	circulation.h
	Push-relabel algorithm for finding a feasible circulation.
file	edmonds_karp.h
	Implementation of the Edmonds-Karp algorithm.
file	preflow.h
	Implementation of the preflow algorithm.