RhodeCode

 *

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

///\ingroup demos

///\file

///\brief Argument parser demo

///

/// This example shows how the argument parser can be used.

///

/// \include arg_parser_demo.cc

#include <lemon/arg_parser.h>

using namespace lemon;

int main(int argc, const char **argv)

{

  // Initialize the argument parser

  ArgParser ap(argc, argv);

  int i;

  std::string s;

  double d = 1.0;

  bool b, nh;

  bool g1, g2, g3;

  // Add a mandatory integer option with storage reference

  ap.refOption("n", "An integer input.", i, true);

  // Add a double option with storage reference (the default value is 1.0)

  ap.refOption("val", "A double input.", d);

  // Add a double option without storage reference (the default value is 3.14)

  ap.doubleOption("val2", "A double input.", 3.14);

  // Set synonym for -val option

  ap.synonym("vals", "val");

  // Add a string option

  ap.refOption("name", "A string input.", s);

  // Add bool options

  ap.refOption("f", "A switch.", b)

    .refOption("nohelp", "", nh)

    .refOption("gra", "Choice A", g1)

    .refOption("grb", "Choice B", g2)

    .refOption("grc", "Choice C", g3);

  // Bundle -gr* options into a group

  ap.optionGroup("gr", "gra")

    .optionGroup("gr", "grb")

    .optionGroup("gr", "grc");

  // Set the group mandatory

  ap.mandatoryGroup("gr");

  // Set the options of the group exclusive (only one option can be given)

  ap.onlyOneGroup("gr");

  // Add non-parsed arguments (e.g. input files)

  ap.other("infile", "The input file.")

    .other("...");

  // Perform the parsing process

  // (in case of any error it terminates the program)

  ap.parse();

  // Check each option if it has been given and print its value

  std::cout << "Parameters of '" << ap.commandName() << "':\n";

  std::cout << "  Value of -n: " << i << std::endl;

  if(ap.given("val")) std::cout << "  Value of -val: " << d << std::endl;

  if(ap.given("val2")) {

    d = ap["val2"];

    std::cout << "  Value of -val2: " << d << std::endl;

  }

  if(ap.given("name")) std::cout << "  Value of -name: " << s << std::endl;

  if(ap.given("f")) std::cout << "  -f is given\n";

  if(ap.given("nohelp")) std::cout << "  Value of -nohelp: " << nh << std::endl;

  if(ap.given("gra")) std::cout << "  -gra is given\n";

  if(ap.given("grb")) std::cout << "  -grb is given\n";

  if(ap.given("grc")) std::cout << "  -grc is given\n";

  switch(ap.files().size()) {

  case 0:

    std::cout << "  No file argument was given.\n";

    break;

  case 1:

    std::cout << "  1 file argument was given. It is:\n";

    break;

  default:

    std::cout << "  "

  }

  for(unsigned int i=0;i<ap.files().size();++i)

    std::cout << "    '" << ap.files()[i] << "'\n";

  return 0;

}

 *

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

/// \ingroup demos

/// \file

/// \brief Demo of the graph drawing function \ref graphToEps()

///

/// This demo program shows examples how to  use the function \ref

/// graphToEps(). It takes no input but simply creates  six

/// <tt>.eps</tt> files demonstrating the capability of \ref

/// graphToEps(), and showing how to draw directed graphs,

/// how to handle parallel egdes, how to change the properties (like

/// color, shape, size, title etc.) of nodes and arcs individually

/// using appropriate \ref maps-page "graph maps".

///

/// \include graph_to_eps_demo.cc

#include<lemon/list_graph.h>

#include<lemon/graph_utils.h>

#include<lemon/graph_to_eps.h>

#include<lemon/math.h>

using namespace std;

using namespace lemon;

int main()

{

  Palette palette;

  Palette paletteW(true);

  // Create a small digraph

  ListDigraph g;

  typedef ListDigraph::Node Node;

  typedef ListDigraph::NodeIt NodeIt;

  typedef ListDigraph::Arc Arc;

  typedef dim2::Point<int> Point;

  Node n1=g.addNode();

  Node n2=g.addNode();

  Node n3=g.addNode();

  Node n4=g.addNode();

  Node n5=g.addNode();

  ListDigraph::NodeMap<Point> coords(g);

  ListDigraph::NodeMap<double> sizes(g);

  ListDigraph::NodeMap<int> colors(g);

  ListDigraph::NodeMap<int> shapes(g);

  ListDigraph::ArcMap<int> acolors(g);

  ListDigraph::ArcMap<int> widths(g);

  coords[n1]=Point(50,50);  sizes[n1]=1; colors[n1]=1; shapes[n1]=0;

  coords[n2]=Point(50,70);  sizes[n2]=2; colors[n2]=2; shapes[n2]=2;

  coords[n3]=Point(70,70);  sizes[n3]=1; colors[n3]=3; shapes[n3]=0;

  coords[n4]=Point(70,50);  sizes[n4]=2; colors[n4]=4; shapes[n4]=1;

  coords[n5]=Point(85,60);  sizes[n5]=3; colors[n5]=5; shapes[n5]=2;

  Arc a;

  a=g.addArc(n1,n2); acolors[a]=0; widths[a]=1;

  a=g.addArc(n2,n3); acolors[a]=0; widths[a]=1;

  a=g.addArc(n3,n5); acolors[a]=0; widths[a]=3;

  a=g.addArc(n5,n4); acolors[a]=0; widths[a]=1;

  a=g.addArc(n4,n1); acolors[a]=0; widths[a]=1;

  a=g.addArc(n2,n4); acolors[a]=1; widths[a]=2;

  a=g.addArc(n3,n4); acolors[a]=2; widths[a]=1;

  IdMap<ListDigraph,Node> id(g);

  // Create five .eps files showing the digraph with different options

  cout << "Create 'graph_to_eps_demo_out_1_pure.eps'" << endl;

  graphToEps(g,"graph_to_eps_demo_out_1_pure.eps").

    coords(coords).

    title("Sample .eps figure").

    copyright("(C) 2003-2008 LEMON Project").

    run();

  cout << "Create 'graph_to_eps_demo_out_2.eps'" << endl;

  graphToEps(g,"graph_to_eps_demo_out_2.eps").

    coords(coords).

    title("Sample .eps figure").

    copyright("(C) 2003-2008 LEMON Project").

    absoluteNodeSizes().absoluteArcWidths().

    nodeScale(2).nodeSizes(sizes).

    nodeShapes(shapes).

    nodeColors(composeMap(palette,colors)).

    arcColors(composeMap(palette,acolors)).

    arcWidthScale(.4).arcWidths(widths).

    nodeTexts(id).nodeTextSize(3).

    run();

  cout << "Create 'graph_to_eps_demo_out_3_arr.eps'" << endl;

  graphToEps(g,"graph_to_eps_demo_out_3_arr.eps").

    title("Sample .eps figure (with arrowheads)").

    copyright("(C) 2003-2008 LEMON Project").

    absoluteNodeSizes().absoluteArcWidths().

    nodeColors(composeMap(palette,colors)).

    coords(coords).

    nodeScale(2).nodeSizes(sizes).

    nodeShapes(shapes).

    arcColors(composeMap(palette,acolors)).

    arcWidthScale(.4).arcWidths(widths).

    nodeTexts(id).nodeTextSize(3).

    drawArrows().arrowWidth(2).arrowLength(2).

    run();

  a=g.addArc(n1,n4); acolors[a]=2; widths[a]=1;

  a=g.addArc(n4,n1); acolors[a]=1; widths[a]=2;

  a=g.addArc(n1,n2); acolors[a]=1; widths[a]=1;

  a=g.addArc(n1,n2); acolors[a]=2; widths[a]=1;

  a=g.addArc(n1,n2); acolors[a]=3; widths[a]=1;

  a=g.addArc(n1,n2); acolors[a]=4; widths[a]=1;

  a=g.addArc(n1,n2); acolors[a]=5; widths[a]=1;

  a=g.addArc(n1,n2); acolors[a]=6; widths[a]=1;

  a=g.addArc(n1,n2); acolors[a]=7; widths[a]=1;

  cout << "Create 'graph_to_eps_demo_out_par.eps'" << endl;

  graphToEps(g,"graph_to_eps_demo_out_par.eps").

    //scale(10).

    title("Sample .eps figure (parallel arcs)").

    copyright("(C) 2003-2008 LEMON Project").

    absoluteNodeSizes().absoluteArcWidths().

    nodeShapes(shapes).

    coords(coords).

    nodeScale(2).nodeSizes(sizes).

    nodeColors(composeMap(palette,colors)).

    arcColors(composeMap(palette,acolors)).

    arcWidthScale(.4).arcWidths(widths).

    nodeTexts(id).nodeTextSize(3).

    enableParallel().parArcDist(1.5).

    run();

  cout << "Create 'graph_to_eps_demo_out_4_par_arr.eps'" << endl;

  graphToEps(g,"graph_to_eps_demo_out_4_par_arr.eps").

    title("Sample .eps figure (parallel arcs and arrowheads)").

    copyright("(C) 2003-2008 LEMON Project").

    absoluteNodeSizes().absoluteArcWidths().

    nodeScale(2).nodeSizes(sizes).

    coords(coords).

    nodeShapes(shapes).

    nodeColors(composeMap(palette,colors)).

    arcColors(composeMap(palette,acolors)).

    arcWidthScale(.3).arcWidths(widths).

    nodeTexts(id).nodeTextSize(3).

    enableParallel().parArcDist(1).

    drawArrows().arrowWidth(1).arrowLength(1).

    run();

  cout << "Create 'graph_to_eps_demo_out_5_par_arr_a4.eps'" << endl;

  graphToEps(g,"graph_to_eps_demo_out_5_par_arr_a4.eps").

    title("Sample .eps figure (fits to A4)").

    copyright("(C) 2003-2008 LEMON Project").

    scaleToA4().

    absoluteNodeSizes().absoluteArcWidths().

    nodeScale(2).nodeSizes(sizes).

    coords(coords).

    nodeShapes(shapes).

    nodeColors(composeMap(palette,colors)).

    arcColors(composeMap(palette,acolors)).

    arcWidthScale(.3).arcWidths(widths).

    nodeTexts(id).nodeTextSize(3).

    enableParallel().parArcDist(1).

    drawArrows().arrowWidth(1).arrowLength(1).

    run();

  // Create an .eps file showing the colors of a default Palette

  ListDigraph h;

  ListDigraph::NodeMap<int> hcolors(h);

  ListDigraph::NodeMap<Point> hcoords(h);

  int cols=int(sqrt(double(palette.size())));

  for(int i=0;i<int(paletteW.size());i++) {

    Node n=h.addNode();

    hcoords[n]=Point(1+i%cols,1+i/cols);

    hcolors[n]=i;

  }

  cout << "Create 'graph_to_eps_demo_out_6_colors.eps'" << endl;

  graphToEps(h,"graph_to_eps_demo_out_6_colors.eps").

    scale(60).

    title("Sample .eps figure (Palette demo)").

    copyright("(C) 2003-2008 LEMON Project").

    coords(hcoords).

    absoluteNodeSizes().absoluteArcWidths().

    nodeScale(.45).

    distantColorNodeTexts().

    nodeTexts(hcolors).nodeTextSize(.6).

    nodeColors(composeMap(paletteW,hcolors)).

    run();

  return 0;

}

 *

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

///\ingroup demos

///\file

///\brief Demonstrating graph input and output

///

/// This program gives an example of how to read and write a digraph

/// \ref lgf-format "LGF" format.

///

/// The \c "digraph.lgf" file:

/// \include digraph.lgf

///

/// And the program which reads it and prints the digraph to the

/// standard output:

/// \include lgf_demo.cc

#include <iostream>

#include <lemon/smart_graph.h>

#include <lemon/lgf_reader.h>

#include <lemon/lgf_writer.h>

using namespace lemon;

int main() {

  SmartDigraph g;

  SmartDigraph::ArcMap<int> cap(g);

  SmartDigraph::Node s, t;

  try {

    digraphReader("digraph.lgf", g). // read the directed graph into g

      arcMap("capacity", cap).       // read the 'capacity' arc map into cap

      node("source", s).             // read 'source' node to s

      node("target", t).             // read 'target' node to t

      run();

  } catch (DataFormatError& error) { // check if there was any error

    std::cerr << "Error: " << error.what() << std::endl;

    return -1;

  }

  std::cout << "A digraph is read from 'digraph.lgf'." << std::endl;

  std::cout << "Number of nodes: " << countNodes(g) << std::endl;

  std::cout << "Number of arcs: " << countArcs(g) << std::endl;

  std::cout << "We can write it to the standard output:" << std::endl;

  digraphWriter(std::cout, g).     // write g to the standard output

    arcMap("capacity", cap).       // write cap into 'capacity'

    node("source", s).             // write s to 'source'

    node("target", t).             // write t to 'target'

    run();

  return 0;

}

 *

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

/*!

\section naming_conv Naming Conventions

In order to make development easier we have made some conventions

according to coding style. These include names of types, classes,

functions, variables, constants and exceptions. If these conventions

are met in one's code then it is easier to read and maintain

it. Please comply with these conventions if you want to contribute

developing LEMON library.

\note When the coding style requires the capitalization of an abbreviation,

only the first letter should be upper case.

\code

XmlReader

\endcode

\warning In some cases we diverge from these rules.

This is primary done because STL uses different naming convention and

in certain cases

it is beneficial to provide STL compatible interface.

\subsection cs-files File Names

The header file names should look like the following.

\code

header_file.h

\endcode

Note that all standard LEMON headers are located in the \c lemon subdirectory,

so you should include them from C++ source like this:

\code

#include <lemon/header_file.h>

\endcode

The source code files use the same style and they have '.cc' extension.

\code

source_code.cc

\endcode

\subsection cs-class Classes and other types

The name of a class or any type should look like the following.

\code

\endcode

\subsection cs-func Methods and other functions

The name of a function should look like the following.

\code

\endcode

\subsection cs-funcs Constants, Macros

The names of constants and macros should look like the following.

\code

\endcode

The names of class and instance member variables and auto variables (=variables used locally in methods) should look like the following.

\code

\endcode

\subsection pri-loc-var Private member variables

Private member variables should start with underscore

\code

_start_with_underscores

\endcode

\subsection cs-excep Exceptions

When writing exceptions please comply the following naming conventions.

\code

ClassNameEndsWithException

\endcode

or

\code

ClassNameEndsWithError

\endcode

\section header-template Template Header File

Each LEMON header file should look like this:

\include template.h

*/

 *

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

/**

\dir demo

\brief A collection of demo application.

This directory contains several simple demo application, mainly

for educational purposes.

*/

/**

\dir doc

\brief Auxiliary (and the whole generated) documentation.

Auxiliary (and the whole generated) documentation.

*/

/**

\dir test

\brief Test programs.

This directory contains several test programs that check the consistency

of the code.

*/

/**

\dir tools

\brief Some useful executables

This directory contains the sources of some useful complete executables.

*/

/**

\dir lemon

\brief Base include directory of LEMON

This is the base directory of lemon includes, so each include file must be

prefixed with this, e.g.

\code

#include<lemon/list_graph.h>

#include<lemon/dijkstra.h>

\endcode

*/

/**

\dir concepts

\brief Concept descriptors and checking classes

This directory contains the concept descriptors and concept checkers. As a user

you typically don't have to deal with these files.

*/

/**

\dir bits

\brief Implementation helper files

This directory contains some helper classes to implement graphs, maps and

files.

*/

 *

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

/**

@defgroup datas Data Structures

This group describes the several data structures implemented in LEMON.

*/

/**

@defgroup graphs Graph Structures

@ingroup datas

\brief Graph structures implemented in LEMON.

The most efficient implementation of diverse applications require the

usage of different physical graph implementations. These differences

appear in the size of graph we require to handle, memory or time usage

limitations or in the set of operations through which the graph can be

accessed.  LEMON provides several physical graph structures to meet

the diverging requirements of the possible users.  In order to save on

running time or on memory usage, some structures may fail to provide

some graph features like arc/edge or node deletion.

arcs have to be hidden or the reverse oriented graph have to be used, then

You are free to use the graph structure that fit your requirements

the best, most graph algorithms and auxiliary data structures can be used

*/

/**

@defgroup semi_adaptors Semi-Adaptor Classes for Graphs

@ingroup graphs

\brief Graph types between real graphs and graph adaptors.

This group describes some graph types between real graphs and graph adaptors.

On the other hand they are not light-weight structures as the adaptors.

*/

/**

@ingroup datas

\brief Map structures implemented in LEMON.

This group describes the map structures implemented in LEMON.

LEMON provides several special purpose maps that e.g. combine

new maps from existing ones.

*/

/**

@ingroup maps

\brief Special graph-related maps.

This group describes maps that are specifically designed to assign

values to the nodes and arcs of graphs.

*/

/**

\defgroup map_adaptors Map Adaptors

\ingroup maps

\brief Tools to create new maps from existing ones

This group describes map adaptors that are used to create "implicit"

maps from other maps.

Most of them are \ref lemon::concepts::ReadMap "read-only maps".

They can make arithmetic and logical operations between one or two maps

(negation, shifting, addition, multiplication, logical 'and', 'or',

'not' etc.) or e.g. convert a map to another one of different Value type.

The typical usage of this classes is passing implicit maps to

algorithms.  If a function type algorithm is called then the function

type map adaptors can be used comfortable. For example let's see the

usage of map adaptors with the \c digraphToEps() function.

\code

  Color nodeColor(int deg) {

    if (deg >= 2) {

      return Color(0.5, 0.0, 0.5);

    } else if (deg == 1) {

      return Color(1.0, 0.5, 1.0);

    } else {

      return Color(0.0, 0.0, 0.0);

    }

  }

  Digraph::NodeMap<int> degree_map(graph);

  digraphToEps(graph, "graph.eps")

    .coords(coords).scaleToA4().undirected()

    .nodeColors(composeMap(functorToMap(nodeColor), degree_map))

    .run();

The \c functorToMap() function makes an \c int to \c Color map from the

\e nodeColor() function. The \c composeMap() compose the \e degree_map

and the previously created map. The composed map is a proper function to

get the color of each node.

The usage with class type algorithms is little bit harder. In this

case the function type map adaptors can not be used, because the

function map adaptors give back temporary objects.

\code

  Digraph graph;

  typedef Digraph::ArcMap<double> DoubleArcMap;

  DoubleArcMap length(graph);

  DoubleArcMap speed(graph);

  typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;

  TimeMap time(length, speed);

  Dijkstra<Digraph, TimeMap> dijkstra(graph, time);

  dijkstra.run(source, target);

\endcode

We have a length map and a maximum speed map on the arcs of a digraph.

The minimum time to pass the arc can be calculated as the division of

the two maps which can be done implicitly with the \c DivMap template

class. We use the implicit minimum time map as the length map of the

\c Dijkstra algorithm.

*/

/**

@ingroup datas

\brief Two dimensional data storages implemented in LEMON.

This group describes two dimensional data storages implemented in LEMON.

*/

/**

@defgroup paths Path Structures

@ingroup datas

\brief Path structures implemented in LEMON.

This group describes the path structures implemented in LEMON.

LEMON provides flexible data structures to work with paths.

All of them have similar interfaces and they can be copied easily with

assignment operators and copy constructors. This makes it easy and

efficient to have e.g. the Dijkstra algorithm to store its result in

any kind of path structure.

\sa lemon::concepts::Path

*/

/**

@defgroup auxdat Auxiliary Data Structures

@ingroup datas

\brief Auxiliary data structures implemented in LEMON.

This group describes some data structures implemented in LEMON in

order to make it easier to implement combinatorial algorithms.

*/

/**

@defgroup algs Algorithms

\brief This group describes the several algorithms

implemented in LEMON.

This group describes the several algorithms

implemented in LEMON.

*/

/**

@defgroup search Graph Search

@ingroup algs

\brief Common graph search algorithms.

Breadth-first search (Bfs) and Depth-first search (Dfs).

*/

/**

@defgroup shortest_path Shortest Path algorithms

@ingroup algs

\brief Algorithms for finding shortest paths.

This group describes the algorithms for finding shortest paths in graphs.

*/

\brief Algorithms for finding maximum flows.

This group describes the algorithms for finding maximum flows and

feasible circulations.

The maximum flow problem is to find a flow between a single source and

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

directed graph, an \f$c_a:A\rightarrow\mathbf{R}^+_0\f$ capacity

function and given \f$s, t \in V\f$ source and target node. The

maximum flow is the \f$f_a\f$ solution of the next optimization problem:

\f[ 0 \le f_a \le c_a \f]

\f[ \sum_{v\in\delta^{-}(u)}f_{vu}=\sum_{v\in\delta^{+}(u)}f_{uv} \qquad \forall u \in V \setminus \{s,t\}\f]

\f[ \max \sum_{v\in\delta^{+}(s)}f_{uv} - \sum_{v\in\delta^{-}(s)}f_{vu}\f]

LEMON contains several algorithms for solving maximum flow problems:

- \ref lemon::Preflow "Goldberg's Preflow algorithm"

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

- \ref lemon::GoldbergTarjan "Preflow algorithm with dynamic trees"

In most cases the \ref lemon::Preflow "Preflow" algorithm provides the

fastest method to compute the maximum flow. All impelementations

provides functions to query the minimum cut, which is the dual linear

programming problem of the maximum flow.

*/

/**

@defgroup min_cost_flow Minimum Cost Flow algorithms

@ingroup algs

\brief Algorithms for finding minimum cost flows and circulations.

This group describes the algorithms for finding minimum cost flows and

*/

/**

\brief Algorithms for finding minimum cut in graphs.

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

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

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

outgoing arcs. Formally, there is \f$G=(V,A)\f$ directed graph, an

\f$c_a:A\rightarrow\mathbf{R}^+_0\f$ 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}c_{uv}\f]

LEMON contains several algorithms related to minimum cut problems:

- \ref lemon::HaoOrlin "Hao-Orlin algorithm" to calculate minimum cut

- \ref lemon::NagamochiIbaraki "Nagamochi-Ibaraki algorithm" to

  calculate minimum cut in undirected graphs

- \ref lemon::GomoryHuTree "Gomory-Hu tree computation" to calculate all

  pairs minimum cut in undirected graphs

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

please see the \ref max_flow "Maximum Flow page".

*/

/**

@defgroup graph_prop Connectivity and other graph properties

@ingroup algs

\brief Algorithms for discovering the graph properties

This group describes the algorithms for discovering the graph properties

like connectivity, bipartiteness, euler property, simplicity etc.

\image html edge_biconnected_components.png

\image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth

*/

/**

@defgroup planar Planarity embedding and drawing

@ingroup algs

\brief Algorithms for planarity checking, embedding and drawing

This group describes the algorithms for planarity checking, embedding and drawing.

\image html planar.png

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

*/

/**

@ingroup algs

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

This group contains algorithm objects and functions to calculate

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

finding a subset of the arcs which does not shares common endpoints.

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 the finding maximum cardinality, maximum weight or minimum cost

matching. The search can be constrained to find perfect or

maximum cardinality matching.

Lemon contains the next algorithms:

  bipartite graphs

  and maximum weighted bipartite matching in bipartite graph

  matching in bipartite graph

- \ref lemon::MaxMatching "MaxMatching" Edmond's blossom shrinking algorithm

  for calculate maximum cardinality matching in general graph

- \ref lemon::MaxWeightedMatching "MaxWeightedMatching" Edmond's blossom

  shrinking algorithm for calculate maximum weighted matching in general

  graph

- \ref lemon::MaxWeightedPerfectMatching "MaxWeightedPerfectMatching"

  Edmond's blossom shrinking algorithm for calculate maximum weighted

  perfect matching in general graph

\image html bipartite_matching.png

\image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth

*/

/**

@defgroup spantree Minimum Spanning Tree algorithms

@ingroup algs

\brief Algorithms for finding a minimum cost spanning tree in a graph.

This group describes the algorithms for finding a minimum cost spanning

tree in a graph

*/

/**

@defgroup auxalg Auxiliary algorithms

@ingroup algs

\brief Auxiliary algorithms implemented in LEMON.

This group describes some algorithms implemented in LEMON

in order to make it easier to implement complex algorithms.

*/

/**

@defgroup approx Approximation algorithms

\brief Approximation algorithms.

This group describes the approximation and heuristic algorithms

implemented in LEMON.

*/

/**

@defgroup gen_opt_group General Optimization Tools

\brief This group describes some general optimization frameworks

implemented in LEMON.

This group describes 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 describes Lp and Mip solver interfaces for LEMON. The

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

interface.

*/

@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.

*/

/**