RhodeCode

   but the digraph should not contain directed cycles with negative total

   length.

 - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms

   for solving the \e all-pairs \e shortest \e paths \e problem when arc

   lenghts can be either positive or negative, but the digraph should

   not contain directed cycles with negative total length.

 - \ref Suurballe A successive shortest path algorithm for finding

   arc-disjoint paths between two nodes having minimum total length.

*/

/**

@defgroup spantree Minimum Spanning Tree Algorithms

@ingroup algs

\brief Algorithms for finding minimum cost spanning trees and arborescences.

This group contains the algorithms for finding minimum cost spanning

trees and arborescences \ref clrs01algorithms.

*/

/**

@defgroup max_flow Maximum Flow Algorithms

@ingroup algs

\brief Algorithms for finding maximum flows.

This group contains the algorithms for finding maximum flows and

feasible circulations \ref clrs01algorithms, \ref amo93networkflows.

The \e maximum \e flow \e problem is to find a flow of maximum value between

a single source and a single target. Formally, there is a \f$G=(V,A)\f$

digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and

\f$s, t \in V\f$ source and target nodes.

A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the

following optimization problem.

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

\f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)

    \quad \forall u\in V\setminus\{s,t\} \f]

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

LEMON contains several algorithms for solving maximum flow problems:

- \ref EdmondsKarp Edmonds-Karp algorithm

  \ref edmondskarp72theoretical.

- \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm

  \ref goldberg88newapproach.

- \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees

  \ref dinic70algorithm, \ref sleator83dynamic.

- \ref GoldbergTarjan !Preflow push-relabel algorithm with dynamic trees

  \ref goldberg88newapproach, \ref sleator83dynamic.

In most cases the \ref Preflow algorithm provides the

fastest method for computing a maximum flow. All implementations

also provide functions to query the minimum cut, which is the dual

problem of maximum flow.

\ref 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 \ref Circulation.

*/

/**

@defgroup min_cost_flow_algs Minimum Cost Flow Algorithms

@ingroup algs

\brief Algorithms for finding minimum cost flows and circulations.

This group contains the algorithms for finding minimum cost flows and

circulations \ref amo93networkflows. For more information about this

problem and its dual solution, see \ref min_cost_flow

"Minimum Cost Flow Problem".

LEMON contains several algorithms for this problem.

 - \ref NetworkSimplex Primal Network Simplex algorithm with various

   pivot strategies \ref dantzig63linearprog, \ref kellyoneill91netsimplex.

 - \ref CostScaling Cost Scaling algorithm based on push/augment and

   relabel operations \ref goldberg90approximation, \ref goldberg97efficient,

   \ref bunnagel98efficient.

 - \ref CapacityScaling Capacity Scaling algorithm based on the successive

   shortest path method \ref edmondskarp72theoretical.

 - \ref CycleCanceling Cycle-Canceling algorithms, two of which are

   strongly polynomial \ref klein67primal, \ref goldberg89cyclecanceling.

In general NetworkSimplex is the most efficient implementation,

but in special cases other algorithms could be faster.

For example, if the total supply and/or capacities are rather small,

CapacityScaling is usually the fastest algorithm (without effective scaling).

*/

/**

@defgroup min_cut Minimum Cut Algorithms

@ingroup algs

\brief Algorithms for finding minimum cut in graphs.

This group contains the algorithms for finding minimum cut in graphs.

The \e minimum \e cut \e problem is to find a non-empty and non-complete

\f$X\f$ subset of the nodes with minimum overall capacity on

outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a

\f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum

cut is the \f$X\f$ solution of the next optimization problem:

\f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}

    \sum_{uv\in A: u\in X, v\not\in X}cap(uv) \f]

LEMON contains several algorithms related to minimum cut problems:

- \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut

  in directed graphs.

- \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for

  calculating minimum cut in undirected graphs.

- \ref GomoryHu "Gomory-Hu tree computation" for calculating

  all-pairs minimum cut in undirected graphs.

If you want to find minimum cut just between two distinict nodes,

see the \ref max_flow "maximum flow problem".

*/

/**

@defgroup min_mean_cycle Minimum Mean Cycle Algorithms

@ingroup algs

\brief Algorithms for finding minimum mean cycles.

This group contains the algorithms for finding minimum mean cycles

\ref clrs01algorithms, \ref amo93networkflows.

The \e minimum \e mean \e cycle \e problem is to find a directed cycle

of minimum mean length (cost) in a digraph.

The mean length of a cycle is the average length of its arcs, i.e. the

ratio between the total length of the cycle and the number of arcs on it.

This problem has an important connection to \e conservative \e length

\e functions, too. A length function on the arcs of a digraph is called

conservative if and only if there is no directed cycle of negative total

length. For an arbitrary length function, the negative of the minimum

cycle mean is the smallest \f$\epsilon\f$ value so that increasing the

arc lengths uniformly by \f$\epsilon\f$ results in a conservative length

function.

LEMON contains three algorithms for solving the minimum mean cycle problem:

- \ref Karp "Karp"'s original algorithm \ref amo93networkflows,

  \ref dasdan98minmeancycle.

- \ref HartmannOrlin "Hartmann-Orlin"'s algorithm, which is an improved

  version of Karp's algorithm \ref dasdan98minmeancycle.

- \ref Howard "Howard"'s policy iteration algorithm

  \ref dasdan98minmeancycle.

In practice, the Howard algorithm proved to be by far the most efficient

one, though the best known theoretical bound on its running time is

exponential.

Both Karp and HartmannOrlin algorithms run in time O(ne) and use space

O(n<sup>2</sup>+e), but the latter one is typically faster due to the

applied early termination scheme.

*/

/**

@defgroup matching Matching Algorithms

@ingroup algs

\brief Algorithms for finding matchings in graphs and bipartite graphs.

This group contains the algorithms for calculating

matchings in graphs and bipartite graphs. The general matching problem is

finding a subset of the edges for which each node has at most one incident

edge.

There are several different algorithms for calculate matchings in

graphs.  The matching problems in bipartite graphs are generally

easier than in general graphs. The goal of the matching optimization

can be finding maximum cardinality, maximum weight or minimum cost

matching. The search can be constrained to find perfect or

maximum cardinality matching.

The matching algorithms implemented in LEMON:

- \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm

  for calculating maximum cardinality matching in bipartite graphs.

- \ref PrBipartiteMatching Push-relabel algorithm

  for calculating maximum cardinality matching in bipartite graphs.

- \ref MaxWeightedBipartiteMatching

  Successive shortest path algorithm for calculating maximum weighted

  matching and maximum weighted bipartite matching in bipartite graphs.

- \ref MinCostMaxBipartiteMatching

  Successive shortest path algorithm for calculating minimum cost maximum

  matching in bipartite graphs.

- \ref MaxMatching Edmond's blossom shrinking algorithm for calculating

  maximum cardinality matching in general graphs.

- \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating

  maximum weighted matching in general graphs.

- \ref MaxWeightedPerfectMatching

  Edmond's blossom shrinking algorithm for calculating maximum weighted

  perfect matching in general graphs.

\image html matching.png

*/

/**

@defgroup graph_properties Connectivity and Other Graph Properties

@ingroup algs

\brief Algorithms for discovering the graph properties

This group contains the algorithms for discovering the graph properties

like connectivity, bipartiteness, euler property, simplicity etc.

\image html connected_components.png

\image latex connected_components.eps "Connected components" width=\textwidth

*/

/**

@defgroup planar Planarity Embedding and Drawing

@ingroup algs

\brief Algorithms for planarity checking, embedding and drawing

This group contains the algorithms for planarity checking,

embedding and drawing.

\image html planar.png

\image latex planar.eps "Plane graph" width=\textwidth

*/

/**

@defgroup approx Approximation Algorithms

@ingroup algs

\brief Approximation algorithms.

This group contains the approximation and heuristic algorithms

implemented in LEMON.

*/

/**

@defgroup auxalg Auxiliary Algorithms

@ingroup algs

\brief Auxiliary algorithms implemented in LEMON.

This group contains some algorithms implemented in LEMON

in order to make it easier to implement complex algorithms.

*/

/**

@defgroup gen_opt_group General Optimization Tools

\brief This group contains some general optimization frameworks

implemented in LEMON.

This group contains some general optimization frameworks

implemented in LEMON.

*/

/**

@defgroup lp_group LP and MIP Solvers

@ingroup gen_opt_group

\brief LP and MIP solver interfaces for LEMON.

This group contains LP and MIP solver interfaces for LEMON.

Various LP solvers could be used in the same manner with this

high-level interface.

The currently supported solvers are \ref glpk, \ref clp, \ref cbc,

\ref cplex, \ref soplex.

*/

/**

@defgroup lp_utils Tools for Lp and Mip Solvers

@ingroup lp_group

\brief Helper tools to the Lp and Mip solvers.

This group adds some helper tools to general optimization framework

implemented in LEMON.

*/

/**

@defgroup metah Metaheuristics

@ingroup gen_opt_group

\brief Metaheuristics for LEMON library.

This group contains some metaheuristic optimization tools.

*/

/**

@defgroup utils Tools and Utilities

\brief Tools and utilities for programming in LEMON

Tools and utilities for programming in LEMON.

*/

/**

@defgroup gutils Basic Graph Utilities

@ingroup utils

\brief Simple basic graph utilities.

This group contains some simple basic graph utilities.

*/

/**

@defgroup misc Miscellaneous Tools

@ingroup utils

\brief Tools for development, debugging and testing.

This group contains several useful tools for development,

debugging and testing.

*/

/**

@defgroup timecount Time Measuring and Counting

@ingroup misc

\brief Simple tools for measuring the performance of algorithms.

This group contains simple tools for measuring the performance

of algorithms.

*/

/**

@defgroup exceptions Exceptions

@ingroup utils

\brief Exceptions defined in LEMON.

This group contains the exceptions defined in LEMON.

*/

/**

@defgroup io_group Input-Output

\brief Graph Input-Output methods

This group contains the tools for importing and exporting graphs

and graph related data. Now it supports the \ref lgf-format

"LEMON Graph Format", the \c DIMACS format and the encapsulated

postscript (EPS) format.

*/

/**

@defgroup lemon_io LEMON Graph Format

@ingroup io_group

\brief Reading and writing LEMON Graph Format.

This group contains methods for reading and writing

\ref lgf-format "LEMON Graph Format".

*/

/**

@defgroup eps_io Postscript Exporting

@ingroup io_group

\brief General \c EPS drawer and graph exporter

This group contains general \c EPS drawing methods and special

graph exporting tools.

*/

/**

@defgroup dimacs_group DIMACS Format

@ingroup io_group

\brief Read and write files in DIMACS format

Tools to read a digraph from or write it to a file in DIMACS format data.

*/

/**

@defgroup nauty_group NAUTY Format

@ingroup io_group

\brief Read \e Nauty format

Tool to read graphs from \e Nauty format data.

*/

/**

@defgroup concept Concepts

\brief Skeleton classes and concept checking classes

This group contains the data/algorithm skeletons and concept checking

classes implemented in LEMON.

The purpose of the classes in this group is fourfold.

- These classes contain the documentations of the %concepts. In order

  to avoid document multiplications, an implementation of a concept

  simply refers to the corresponding concept class.

- These classes declare every functions, <tt>typedef</tt>s etc. an

  implementation of the %concepts should provide, however completely

  without implementations and real data structures behind the

  interface. On the other hand they should provide nothing else. All

  the algorithms working on a data structure meeting a certain concept

  should compile with these classes. (Though it will not run properly,

  of course.) In this way it is easily to check if an algorithm

  doesn't use any extra feature of a certain implementation.

- The concept descriptor classes also provide a <em>checker class</em>

  that makes it possible to check whether a certain implementation of a