1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
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2 | * |
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3 | * This file is a part of LEMON, a generic C++ optimization library. |
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4 | * |
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5 | * Copyright (C) 2003-2009 |
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6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
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8 | * |
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9 | * Permission to use, modify and distribute this software is granted |
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10 | * provided that this copyright notice appears in all copies. For |
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11 | * precise terms see the accompanying LICENSE file. |
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12 | * |
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13 | * This software is provided "AS IS" with no warranty of any kind, |
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14 | * express or implied, and with no claim as to its suitability for any |
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15 | * purpose. |
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16 | * |
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17 | */ |
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18 | |
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19 | namespace lemon { |
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20 | |
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21 | /** |
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22 | @defgroup datas Data Structures |
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23 | This group contains the several data structures implemented in LEMON. |
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24 | */ |
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25 | |
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26 | /** |
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27 | @defgroup graphs Graph Structures |
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28 | @ingroup datas |
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29 | \brief Graph structures implemented in LEMON. |
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30 | |
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31 | The implementation of combinatorial algorithms heavily relies on |
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32 | efficient graph implementations. LEMON offers data structures which are |
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33 | planned to be easily used in an experimental phase of implementation studies, |
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34 | and thereafter the program code can be made efficient by small modifications. |
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35 | |
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36 | The most efficient implementation of diverse applications require the |
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37 | usage of different physical graph implementations. These differences |
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38 | appear in the size of graph we require to handle, memory or time usage |
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39 | limitations or in the set of operations through which the graph can be |
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40 | accessed. LEMON provides several physical graph structures to meet |
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41 | the diverging requirements of the possible users. In order to save on |
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42 | running time or on memory usage, some structures may fail to provide |
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43 | some graph features like arc/edge or node deletion. |
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44 | |
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45 | Alteration of standard containers need a very limited number of |
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46 | operations, these together satisfy the everyday requirements. |
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47 | In the case of graph structures, different operations are needed which do |
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48 | not alter the physical graph, but gives another view. If some nodes or |
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49 | arcs have to be hidden or the reverse oriented graph have to be used, then |
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50 | this is the case. It also may happen that in a flow implementation |
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51 | the residual graph can be accessed by another algorithm, or a node-set |
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52 | is to be shrunk for another algorithm. |
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53 | LEMON also provides a variety of graphs for these requirements called |
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54 | \ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only |
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55 | in conjunction with other graph representations. |
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56 | |
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57 | You are free to use the graph structure that fit your requirements |
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58 | the best, most graph algorithms and auxiliary data structures can be used |
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59 | with any graph structure. |
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60 | |
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61 | <b>See also:</b> \ref graph_concepts "Graph Structure Concepts". |
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62 | */ |
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63 | |
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64 | /** |
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65 | @defgroup graph_adaptors Adaptor Classes for Graphs |
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66 | @ingroup graphs |
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67 | \brief Adaptor classes for digraphs and graphs |
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68 | |
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69 | This group contains several useful adaptor classes for digraphs and graphs. |
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70 | |
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71 | The main parts of LEMON are the different graph structures, generic |
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72 | graph algorithms, graph concepts, which couple them, and graph |
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73 | adaptors. While the previous notions are more or less clear, the |
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74 | latter one needs further explanation. Graph adaptors are graph classes |
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75 | which serve for considering graph structures in different ways. |
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76 | |
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77 | A short example makes this much clearer. Suppose that we have an |
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78 | instance \c g of a directed graph type, say ListDigraph and an algorithm |
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79 | \code |
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80 | template <typename Digraph> |
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81 | int algorithm(const Digraph&); |
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82 | \endcode |
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83 | is needed to run on the reverse oriented graph. It may be expensive |
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84 | (in time or in memory usage) to copy \c g with the reversed |
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85 | arcs. In this case, an adaptor class is used, which (according |
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86 | to LEMON \ref concepts::Digraph "digraph concepts") works as a digraph. |
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87 | The adaptor uses the original digraph structure and digraph operations when |
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88 | methods of the reversed oriented graph are called. This means that the adaptor |
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89 | have minor memory usage, and do not perform sophisticated algorithmic |
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90 | actions. The purpose of it is to give a tool for the cases when a |
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91 | graph have to be used in a specific alteration. If this alteration is |
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92 | obtained by a usual construction like filtering the node or the arc set or |
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93 | considering a new orientation, then an adaptor is worthwhile to use. |
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94 | To come back to the reverse oriented graph, in this situation |
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95 | \code |
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96 | template<typename Digraph> class ReverseDigraph; |
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97 | \endcode |
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98 | template class can be used. The code looks as follows |
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99 | \code |
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100 | ListDigraph g; |
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101 | ReverseDigraph<ListDigraph> rg(g); |
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102 | int result = algorithm(rg); |
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103 | \endcode |
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104 | During running the algorithm, the original digraph \c g is untouched. |
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105 | This techniques give rise to an elegant code, and based on stable |
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106 | graph adaptors, complex algorithms can be implemented easily. |
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107 | |
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108 | In flow, circulation and matching problems, the residual |
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109 | graph is of particular importance. Combining an adaptor implementing |
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110 | this with shortest path algorithms or minimum mean cycle algorithms, |
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111 | a range of weighted and cardinality optimization algorithms can be |
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112 | obtained. For other examples, the interested user is referred to the |
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113 | detailed documentation of particular adaptors. |
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114 | |
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115 | The behavior of graph adaptors can be very different. Some of them keep |
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116 | capabilities of the original graph while in other cases this would be |
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117 | meaningless. This means that the concepts that they meet depend |
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118 | on the graph adaptor, and the wrapped graph. |
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119 | For example, if an arc of a reversed digraph is deleted, this is carried |
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120 | out by deleting the corresponding arc of the original digraph, thus the |
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121 | adaptor modifies the original digraph. |
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122 | However in case of a residual digraph, this operation has no sense. |
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123 | |
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124 | Let us stand one more example here to simplify your work. |
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125 | ReverseDigraph has constructor |
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126 | \code |
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127 | ReverseDigraph(Digraph& digraph); |
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128 | \endcode |
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129 | This means that in a situation, when a <tt>const %ListDigraph&</tt> |
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130 | reference to a graph is given, then it have to be instantiated with |
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131 | <tt>Digraph=const %ListDigraph</tt>. |
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132 | \code |
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133 | int algorithm1(const ListDigraph& g) { |
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134 | ReverseDigraph<const ListDigraph> rg(g); |
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135 | return algorithm2(rg); |
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136 | } |
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137 | \endcode |
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138 | */ |
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139 | |
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140 | /** |
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141 | @defgroup maps Maps |
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142 | @ingroup datas |
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143 | \brief Map structures implemented in LEMON. |
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144 | |
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145 | This group contains the map structures implemented in LEMON. |
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146 | |
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147 | LEMON provides several special purpose maps and map adaptors that e.g. combine |
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148 | new maps from existing ones. |
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149 | |
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150 | <b>See also:</b> \ref map_concepts "Map Concepts". |
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151 | */ |
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152 | |
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153 | /** |
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154 | @defgroup graph_maps Graph Maps |
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155 | @ingroup maps |
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156 | \brief Special graph-related maps. |
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157 | |
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158 | This group contains maps that are specifically designed to assign |
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159 | values to the nodes and arcs/edges of graphs. |
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160 | |
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161 | If you are looking for the standard graph maps (\c NodeMap, \c ArcMap, |
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162 | \c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts". |
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163 | */ |
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164 | |
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165 | /** |
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166 | \defgroup map_adaptors Map Adaptors |
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167 | \ingroup maps |
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168 | \brief Tools to create new maps from existing ones |
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169 | |
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170 | This group contains map adaptors that are used to create "implicit" |
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171 | maps from other maps. |
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172 | |
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173 | Most of them are \ref concepts::ReadMap "read-only maps". |
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174 | They can make arithmetic and logical operations between one or two maps |
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175 | (negation, shifting, addition, multiplication, logical 'and', 'or', |
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176 | 'not' etc.) or e.g. convert a map to another one of different Value type. |
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177 | |
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178 | The typical usage of this classes is passing implicit maps to |
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179 | algorithms. If a function type algorithm is called then the function |
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180 | type map adaptors can be used comfortable. For example let's see the |
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181 | usage of map adaptors with the \c graphToEps() function. |
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182 | \code |
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183 | Color nodeColor(int deg) { |
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184 | if (deg >= 2) { |
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185 | return Color(0.5, 0.0, 0.5); |
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186 | } else if (deg == 1) { |
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187 | return Color(1.0, 0.5, 1.0); |
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188 | } else { |
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189 | return Color(0.0, 0.0, 0.0); |
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190 | } |
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191 | } |
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192 | |
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193 | Digraph::NodeMap<int> degree_map(graph); |
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194 | |
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195 | graphToEps(graph, "graph.eps") |
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196 | .coords(coords).scaleToA4().undirected() |
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197 | .nodeColors(composeMap(functorToMap(nodeColor), degree_map)) |
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198 | .run(); |
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199 | \endcode |
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200 | The \c functorToMap() function makes an \c int to \c Color map from the |
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201 | \c nodeColor() function. The \c composeMap() compose the \c degree_map |
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202 | and the previously created map. The composed map is a proper function to |
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203 | get the color of each node. |
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204 | |
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205 | The usage with class type algorithms is little bit harder. In this |
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206 | case the function type map adaptors can not be used, because the |
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207 | function map adaptors give back temporary objects. |
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208 | \code |
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209 | Digraph graph; |
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210 | |
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211 | typedef Digraph::ArcMap<double> DoubleArcMap; |
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212 | DoubleArcMap length(graph); |
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213 | DoubleArcMap speed(graph); |
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214 | |
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215 | typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap; |
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216 | TimeMap time(length, speed); |
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217 | |
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218 | Dijkstra<Digraph, TimeMap> dijkstra(graph, time); |
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219 | dijkstra.run(source, target); |
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220 | \endcode |
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221 | We have a length map and a maximum speed map on the arcs of a digraph. |
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222 | The minimum time to pass the arc can be calculated as the division of |
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223 | the two maps which can be done implicitly with the \c DivMap template |
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224 | class. We use the implicit minimum time map as the length map of the |
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225 | \c Dijkstra algorithm. |
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226 | */ |
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227 | |
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228 | /** |
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229 | @defgroup matrices Matrices |
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230 | @ingroup datas |
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231 | \brief Two dimensional data storages implemented in LEMON. |
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232 | |
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233 | This group contains two dimensional data storages implemented in LEMON. |
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234 | */ |
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235 | |
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236 | /** |
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237 | @defgroup paths Path Structures |
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238 | @ingroup datas |
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239 | \brief %Path structures implemented in LEMON. |
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240 | |
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241 | This group contains the path structures implemented in LEMON. |
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242 | |
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243 | LEMON provides flexible data structures to work with paths. |
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244 | All of them have similar interfaces and they can be copied easily with |
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245 | assignment operators and copy constructors. This makes it easy and |
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246 | efficient to have e.g. the Dijkstra algorithm to store its result in |
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247 | any kind of path structure. |
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248 | |
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249 | \sa lemon::concepts::Path |
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250 | */ |
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251 | |
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252 | /** |
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253 | @defgroup auxdat Auxiliary Data Structures |
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254 | @ingroup datas |
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255 | \brief Auxiliary data structures implemented in LEMON. |
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256 | |
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257 | This group contains some data structures implemented in LEMON in |
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258 | order to make it easier to implement combinatorial algorithms. |
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259 | */ |
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260 | |
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261 | /** |
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262 | @defgroup algs Algorithms |
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263 | \brief This group contains the several algorithms |
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264 | implemented in LEMON. |
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265 | |
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266 | This group contains the several algorithms |
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267 | implemented in LEMON. |
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268 | */ |
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269 | |
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270 | /** |
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271 | @defgroup search Graph Search |
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272 | @ingroup algs |
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273 | \brief Common graph search algorithms. |
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274 | |
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275 | This group contains the common graph search algorithms, namely |
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276 | \e breadth-first \e search (BFS) and \e depth-first \e search (DFS). |
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277 | */ |
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278 | |
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279 | /** |
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280 | @defgroup shortest_path Shortest Path Algorithms |
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281 | @ingroup algs |
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282 | \brief Algorithms for finding shortest paths. |
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283 | |
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284 | This group contains the algorithms for finding shortest paths in digraphs. |
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285 | |
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286 | - \ref Dijkstra algorithm for finding shortest paths from a source node |
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287 | when all arc lengths are non-negative. |
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288 | - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths |
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289 | from a source node when arc lenghts can be either positive or negative, |
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290 | but the digraph should not contain directed cycles with negative total |
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291 | length. |
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292 | - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms |
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293 | for solving the \e all-pairs \e shortest \e paths \e problem when arc |
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294 | lenghts can be either positive or negative, but the digraph should |
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295 | not contain directed cycles with negative total length. |
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296 | - \ref Suurballe A successive shortest path algorithm for finding |
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297 | arc-disjoint paths between two nodes having minimum total length. |
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298 | */ |
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299 | |
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300 | /** |
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301 | @defgroup max_flow Maximum Flow Algorithms |
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302 | @ingroup algs |
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303 | \brief Algorithms for finding maximum flows. |
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304 | |
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305 | This group contains the algorithms for finding maximum flows and |
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306 | feasible circulations. |
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307 | |
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308 | The \e maximum \e flow \e problem is to find a flow of maximum value between |
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309 | a single source and a single target. Formally, there is a \f$G=(V,A)\f$ |
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310 | digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and |
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311 | \f$s, t \in V\f$ source and target nodes. |
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312 | A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the |
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313 | following optimization problem. |
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314 | |
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315 | \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f] |
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316 | \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu) |
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317 | \quad \forall u\in V\setminus\{s,t\} \f] |
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318 | \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f] |
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319 | |
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320 | LEMON contains several algorithms for solving maximum flow problems: |
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321 | - \ref EdmondsKarp Edmonds-Karp algorithm. |
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322 | - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm. |
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323 | - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees. |
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324 | - \ref GoldbergTarjan Preflow push-relabel algorithm with dynamic trees. |
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325 | |
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326 | In most cases the \ref Preflow "Preflow" algorithm provides the |
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327 | fastest method for computing a maximum flow. All implementations |
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328 | also provide functions to query the minimum cut, which is the dual |
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329 | problem of maximum flow. |
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330 | |
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331 | \ref Circulation is a preflow push-relabel algorithm implemented directly |
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332 | for finding feasible circulations, which is a somewhat different problem, |
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333 | but it is strongly related to maximum flow. |
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334 | For more information, see \ref Circulation. |
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335 | */ |
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336 | |
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337 | /** |
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338 | @defgroup min_cost_flow Minimum Cost Flow Algorithms |
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339 | @ingroup algs |
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340 | |
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341 | \brief Algorithms for finding minimum cost flows and circulations. |
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342 | |
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343 | This group contains the algorithms for finding minimum cost flows and |
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344 | circulations. |
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345 | |
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346 | The \e minimum \e cost \e flow \e problem is to find a feasible flow of |
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347 | minimum total cost from a set of supply nodes to a set of demand nodes |
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348 | in a network with capacity constraints (lower and upper bounds) |
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349 | and arc costs. |
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350 | Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{Z}\f$, |
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351 | \f$upper: A\rightarrow\mathbf{Z}\cup\{+\infty\}\f$ denote the lower and |
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352 | upper bounds for the flow values on the arcs, for which |
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353 | \f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$, |
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354 | \f$cost: A\rightarrow\mathbf{Z}\f$ denotes the cost per unit flow |
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355 | on the arcs and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the |
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356 | signed supply values of the nodes. |
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357 | If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$ |
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358 | supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with |
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359 | \f$-sup(u)\f$ demand. |
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360 | A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}\f$ solution |
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361 | of the following optimization problem. |
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362 | |
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363 | \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] |
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364 | \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq |
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365 | sup(u) \quad \forall u\in V \f] |
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366 | \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] |
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367 | |
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368 | The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be |
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369 | zero or negative in order to have a feasible solution (since the sum |
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370 | of the expressions on the left-hand side of the inequalities is zero). |
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371 | It means that the total demand must be greater or equal to the total |
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372 | supply and all the supplies have to be carried out from the supply nodes, |
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373 | but there could be demands that are not satisfied. |
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374 | If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand |
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375 | constraints have to be satisfied with equality, i.e. all demands |
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376 | have to be satisfied and all supplies have to be used. |
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377 | |
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378 | If you need the opposite inequalities in the supply/demand constraints |
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379 | (i.e. the total demand is less than the total supply and all the demands |
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380 | have to be satisfied while there could be supplies that are not used), |
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381 | then you could easily transform the problem to the above form by reversing |
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382 | the direction of the arcs and taking the negative of the supply values |
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383 | (e.g. using \ref ReverseDigraph and \ref NegMap adaptors). |
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384 | However \ref NetworkSimplex algorithm also supports this form directly |
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385 | for the sake of convenience. |
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386 | |
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387 | A feasible solution for this problem can be found using \ref Circulation. |
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388 | |
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389 | Note that the above formulation is actually more general than the usual |
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390 | definition of the minimum cost flow problem, in which strict equalities |
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391 | are required in the supply/demand contraints, i.e. |
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392 | |
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393 | \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) = |
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394 | sup(u) \quad \forall u\in V. \f] |
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395 | |
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396 | However if the sum of the supply values is zero, then these two problems |
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397 | are equivalent. So if you need the equality form, you have to ensure this |
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398 | additional contraint for the algorithms. |
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399 | |
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400 | The dual solution of the minimum cost flow problem is represented by node |
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401 | potentials \f$\pi: V\rightarrow\mathbf{Z}\f$. |
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402 | An \f$f: A\rightarrow\mathbf{Z}\f$ feasible solution of the problem |
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403 | is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{Z}\f$ |
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404 | node potentials the following \e complementary \e slackness optimality |
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405 | conditions hold. |
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406 | |
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407 | - For all \f$uv\in A\f$ arcs: |
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408 | - if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$; |
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409 | - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$; |
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410 | - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$. |
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411 | - For all \f$u\in V\f$ nodes: |
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412 | - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$, |
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413 | then \f$\pi(u)=0\f$. |
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414 | |
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415 | Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc |
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416 | \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e. |
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417 | \f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f] |
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418 | |
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419 | All algorithms provide dual solution (node potentials) as well, |
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420 | if an optimal flow is found. |
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421 | |
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422 | LEMON contains several algorithms for solving minimum cost flow problems. |
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423 | - \ref NetworkSimplex Primal Network Simplex algorithm with various |
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424 | pivot strategies. |
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425 | - \ref CostScaling Push-Relabel and Augment-Relabel algorithms based on |
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426 | cost scaling. |
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427 | - \ref CapacityScaling Successive Shortest %Path algorithm with optional |
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428 | capacity scaling. |
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429 | - \ref CancelAndTighten The Cancel and Tighten algorithm. |
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430 | - \ref CycleCanceling Cycle-Canceling algorithms. |
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431 | |
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432 | Most of these implementations support the general inequality form of the |
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433 | minimum cost flow problem, but CancelAndTighten and CycleCanceling |
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434 | only support the equality form due to the primal method they use. |
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435 | |
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436 | In general NetworkSimplex is the most efficient implementation, |
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437 | but in special cases other algorithms could be faster. |
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438 | For example, if the total supply and/or capacities are rather small, |
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439 | CapacityScaling is usually the fastest algorithm (without effective scaling). |
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440 | */ |
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441 | |
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442 | /** |
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443 | @defgroup min_cut Minimum Cut Algorithms |
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444 | @ingroup algs |
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445 | |
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446 | \brief Algorithms for finding minimum cut in graphs. |
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447 | |
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448 | This group contains the algorithms for finding minimum cut in graphs. |
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449 | |
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450 | The \e minimum \e cut \e problem is to find a non-empty and non-complete |
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451 | \f$X\f$ subset of the nodes with minimum overall capacity on |
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452 | outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a |
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453 | \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum |
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454 | cut is the \f$X\f$ solution of the next optimization problem: |
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455 | |
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456 | \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}} |
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457 | \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f] |
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458 | |
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459 | LEMON contains several algorithms related to minimum cut problems: |
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460 | |
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461 | - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut |
---|
462 | in directed graphs. |
---|
463 | - \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for |
---|
464 | calculating minimum cut in undirected graphs. |
---|
465 | - \ref GomoryHu "Gomory-Hu tree computation" for calculating |
---|
466 | all-pairs minimum cut in undirected graphs. |
---|
467 | |
---|
468 | If you want to find minimum cut just between two distinict nodes, |
---|
469 | see the \ref max_flow "maximum flow problem". |
---|
470 | */ |
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471 | |
---|
472 | /** |
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473 | @defgroup graph_properties Connectivity and Other Graph Properties |
---|
474 | @ingroup algs |
---|
475 | \brief Algorithms for discovering the graph properties |
---|
476 | |
---|
477 | This group contains the algorithms for discovering the graph properties |
---|
478 | like connectivity, bipartiteness, euler property, simplicity etc. |
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479 | |
---|
480 | \image html edge_biconnected_components.png |
---|
481 | \image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth |
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482 | */ |
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483 | |
---|
484 | /** |
---|
485 | @defgroup planar Planarity Embedding and Drawing |
---|
486 | @ingroup algs |
---|
487 | \brief Algorithms for planarity checking, embedding and drawing |
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488 | |
---|
489 | This group contains the algorithms for planarity checking, |
---|
490 | embedding and drawing. |
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491 | |
---|
492 | \image html planar.png |
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493 | \image latex planar.eps "Plane graph" width=\textwidth |
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494 | */ |
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495 | |
---|
496 | /** |
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497 | @defgroup matching Matching Algorithms |
---|
498 | @ingroup algs |
---|
499 | \brief Algorithms for finding matchings in graphs and bipartite graphs. |
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500 | |
---|
501 | This group contains the algorithms for calculating |
---|
502 | matchings in graphs and bipartite graphs. The general matching problem is |
---|
503 | finding a subset of the edges for which each node has at most one incident |
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504 | edge. |
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505 | |
---|
506 | There are several different algorithms for calculate matchings in |
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507 | graphs. The matching problems in bipartite graphs are generally |
---|
508 | easier than in general graphs. The goal of the matching optimization |
---|
509 | can be finding maximum cardinality, maximum weight or minimum cost |
---|
510 | matching. The search can be constrained to find perfect or |
---|
511 | maximum cardinality matching. |
---|
512 | |
---|
513 | The matching algorithms implemented in LEMON: |
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514 | - \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm |
---|
515 | for calculating maximum cardinality matching in bipartite graphs. |
---|
516 | - \ref PrBipartiteMatching Push-relabel algorithm |
---|
517 | for calculating maximum cardinality matching in bipartite graphs. |
---|
518 | - \ref MaxWeightedBipartiteMatching |
---|
519 | Successive shortest path algorithm for calculating maximum weighted |
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520 | matching and maximum weighted bipartite matching in bipartite graphs. |
---|
521 | - \ref MinCostMaxBipartiteMatching |
---|
522 | Successive shortest path algorithm for calculating minimum cost maximum |
---|
523 | matching in bipartite graphs. |
---|
524 | - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating |
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525 | maximum cardinality matching in general graphs. |
---|
526 | - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating |
---|
527 | maximum weighted matching in general graphs. |
---|
528 | - \ref MaxWeightedPerfectMatching |
---|
529 | Edmond's blossom shrinking algorithm for calculating maximum weighted |
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530 | perfect matching in general graphs. |
---|
531 | |
---|
532 | \image html bipartite_matching.png |
---|
533 | \image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth |
---|
534 | */ |
---|
535 | |
---|
536 | /** |
---|
537 | @defgroup spantree Minimum Spanning Tree Algorithms |
---|
538 | @ingroup algs |
---|
539 | \brief Algorithms for finding minimum cost spanning trees and arborescences. |
---|
540 | |
---|
541 | This group contains the algorithms for finding minimum cost spanning |
---|
542 | trees and arborescences. |
---|
543 | */ |
---|
544 | |
---|
545 | /** |
---|
546 | @defgroup auxalg Auxiliary Algorithms |
---|
547 | @ingroup algs |
---|
548 | \brief Auxiliary algorithms implemented in LEMON. |
---|
549 | |
---|
550 | This group contains some algorithms implemented in LEMON |
---|
551 | in order to make it easier to implement complex algorithms. |
---|
552 | */ |
---|
553 | |
---|
554 | /** |
---|
555 | @defgroup approx Approximation Algorithms |
---|
556 | @ingroup algs |
---|
557 | \brief Approximation algorithms. |
---|
558 | |
---|
559 | This group contains the approximation and heuristic algorithms |
---|
560 | implemented in LEMON. |
---|
561 | */ |
---|
562 | |
---|
563 | /** |
---|
564 | @defgroup gen_opt_group General Optimization Tools |
---|
565 | \brief This group contains some general optimization frameworks |
---|
566 | implemented in LEMON. |
---|
567 | |
---|
568 | This group contains some general optimization frameworks |
---|
569 | implemented in LEMON. |
---|
570 | */ |
---|
571 | |
---|
572 | /** |
---|
573 | @defgroup lp_group Lp and Mip Solvers |
---|
574 | @ingroup gen_opt_group |
---|
575 | \brief Lp and Mip solver interfaces for LEMON. |
---|
576 | |
---|
577 | This group contains Lp and Mip solver interfaces for LEMON. The |
---|
578 | various LP solvers could be used in the same manner with this |
---|
579 | interface. |
---|
580 | */ |
---|
581 | |
---|
582 | /** |
---|
583 | @defgroup lp_utils Tools for Lp and Mip Solvers |
---|
584 | @ingroup lp_group |
---|
585 | \brief Helper tools to the Lp and Mip solvers. |
---|
586 | |
---|
587 | This group adds some helper tools to general optimization framework |
---|
588 | implemented in LEMON. |
---|
589 | */ |
---|
590 | |
---|
591 | /** |
---|
592 | @defgroup metah Metaheuristics |
---|
593 | @ingroup gen_opt_group |
---|
594 | \brief Metaheuristics for LEMON library. |
---|
595 | |
---|
596 | This group contains some metaheuristic optimization tools. |
---|
597 | */ |
---|
598 | |
---|
599 | /** |
---|
600 | @defgroup utils Tools and Utilities |
---|
601 | \brief Tools and utilities for programming in LEMON |
---|
602 | |
---|
603 | Tools and utilities for programming in LEMON. |
---|
604 | */ |
---|
605 | |
---|
606 | /** |
---|
607 | @defgroup gutils Basic Graph Utilities |
---|
608 | @ingroup utils |
---|
609 | \brief Simple basic graph utilities. |
---|
610 | |
---|
611 | This group contains some simple basic graph utilities. |
---|
612 | */ |
---|
613 | |
---|
614 | /** |
---|
615 | @defgroup misc Miscellaneous Tools |
---|
616 | @ingroup utils |
---|
617 | \brief Tools for development, debugging and testing. |
---|
618 | |
---|
619 | This group contains several useful tools for development, |
---|
620 | debugging and testing. |
---|
621 | */ |
---|
622 | |
---|
623 | /** |
---|
624 | @defgroup timecount Time Measuring and Counting |
---|
625 | @ingroup misc |
---|
626 | \brief Simple tools for measuring the performance of algorithms. |
---|
627 | |
---|
628 | This group contains simple tools for measuring the performance |
---|
629 | of algorithms. |
---|
630 | */ |
---|
631 | |
---|
632 | /** |
---|
633 | @defgroup exceptions Exceptions |
---|
634 | @ingroup utils |
---|
635 | \brief Exceptions defined in LEMON. |
---|
636 | |
---|
637 | This group contains the exceptions defined in LEMON. |
---|
638 | */ |
---|
639 | |
---|
640 | /** |
---|
641 | @defgroup io_group Input-Output |
---|
642 | \brief Graph Input-Output methods |
---|
643 | |
---|
644 | This group contains the tools for importing and exporting graphs |
---|
645 | and graph related data. Now it supports the \ref lgf-format |
---|
646 | "LEMON Graph Format", the \c DIMACS format and the encapsulated |
---|
647 | postscript (EPS) format. |
---|
648 | */ |
---|
649 | |
---|
650 | /** |
---|
651 | @defgroup lemon_io LEMON Graph Format |
---|
652 | @ingroup io_group |
---|
653 | \brief Reading and writing LEMON Graph Format. |
---|
654 | |
---|
655 | This group contains methods for reading and writing |
---|
656 | \ref lgf-format "LEMON Graph Format". |
---|
657 | */ |
---|
658 | |
---|
659 | /** |
---|
660 | @defgroup eps_io Postscript Exporting |
---|
661 | @ingroup io_group |
---|
662 | \brief General \c EPS drawer and graph exporter |
---|
663 | |
---|
664 | This group contains general \c EPS drawing methods and special |
---|
665 | graph exporting tools. |
---|
666 | */ |
---|
667 | |
---|
668 | /** |
---|
669 | @defgroup dimacs_group DIMACS format |
---|
670 | @ingroup io_group |
---|
671 | \brief Read and write files in DIMACS format |
---|
672 | |
---|
673 | Tools to read a digraph from or write it to a file in DIMACS format data. |
---|
674 | */ |
---|
675 | |
---|
676 | /** |
---|
677 | @defgroup nauty_group NAUTY Format |
---|
678 | @ingroup io_group |
---|
679 | \brief Read \e Nauty format |
---|
680 | |
---|
681 | Tool to read graphs from \e Nauty format data. |
---|
682 | */ |
---|
683 | |
---|
684 | /** |
---|
685 | @defgroup concept Concepts |
---|
686 | \brief Skeleton classes and concept checking classes |
---|
687 | |
---|
688 | This group contains the data/algorithm skeletons and concept checking |
---|
689 | classes implemented in LEMON. |
---|
690 | |
---|
691 | The purpose of the classes in this group is fourfold. |
---|
692 | |
---|
693 | - These classes contain the documentations of the %concepts. In order |
---|
694 | to avoid document multiplications, an implementation of a concept |
---|
695 | simply refers to the corresponding concept class. |
---|
696 | |
---|
697 | - These classes declare every functions, <tt>typedef</tt>s etc. an |
---|
698 | implementation of the %concepts should provide, however completely |
---|
699 | without implementations and real data structures behind the |
---|
700 | interface. On the other hand they should provide nothing else. All |
---|
701 | the algorithms working on a data structure meeting a certain concept |
---|
702 | should compile with these classes. (Though it will not run properly, |
---|
703 | of course.) In this way it is easily to check if an algorithm |
---|
704 | doesn't use any extra feature of a certain implementation. |
---|
705 | |
---|
706 | - The concept descriptor classes also provide a <em>checker class</em> |
---|
707 | that makes it possible to check whether a certain implementation of a |
---|
708 | concept indeed provides all the required features. |
---|
709 | |
---|
710 | - Finally, They can serve as a skeleton of a new implementation of a concept. |
---|
711 | */ |
---|
712 | |
---|
713 | /** |
---|
714 | @defgroup graph_concepts Graph Structure Concepts |
---|
715 | @ingroup concept |
---|
716 | \brief Skeleton and concept checking classes for graph structures |
---|
717 | |
---|
718 | This group contains the skeletons and concept checking classes of LEMON's |
---|
719 | graph structures and helper classes used to implement these. |
---|
720 | */ |
---|
721 | |
---|
722 | /** |
---|
723 | @defgroup map_concepts Map Concepts |
---|
724 | @ingroup concept |
---|
725 | \brief Skeleton and concept checking classes for maps |
---|
726 | |
---|
727 | This group contains the skeletons and concept checking classes of maps. |
---|
728 | */ |
---|
729 | |
---|
730 | /** |
---|
731 | \anchor demoprograms |
---|
732 | |
---|
733 | @defgroup demos Demo Programs |
---|
734 | |
---|
735 | Some demo programs are listed here. Their full source codes can be found in |
---|
736 | the \c demo subdirectory of the source tree. |
---|
737 | |
---|
738 | In order to compile them, use the <tt>make demo</tt> or the |
---|
739 | <tt>make check</tt> commands. |
---|
740 | */ |
---|
741 | |
---|
742 | /** |
---|
743 | @defgroup tools Standalone Utility Applications |
---|
744 | |
---|
745 | Some utility applications are listed here. |
---|
746 | |
---|
747 | The standard compilation procedure (<tt>./configure;make</tt>) will compile |
---|
748 | them, as well. |
---|
749 | */ |
---|
750 | |
---|
751 | } |
---|