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 | \page min_cost_flow Minimum Cost Flow Problem |
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23 | |
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24 | \section mcf_def Definition (GEQ form) |
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25 | |
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26 | The \e minimum \e cost \e flow \e problem is to find a feasible flow of |
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27 | minimum total cost from a set of supply nodes to a set of demand nodes |
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28 | in a network with capacity constraints (lower and upper bounds) |
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29 | and arc costs. |
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30 | |
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31 | Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{R}\f$, |
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32 | \f$upper: A\rightarrow\mathbf{R}\cup\{+\infty\}\f$ denote the lower and |
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33 | upper bounds for the flow values on the arcs, for which |
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34 | \f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$, |
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35 | \f$cost: A\rightarrow\mathbf{R}\f$ denotes the cost per unit flow |
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36 | on the arcs and \f$sup: V\rightarrow\mathbf{R}\f$ denotes the |
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37 | signed supply values of the nodes. |
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38 | If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$ |
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39 | supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with |
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40 | \f$-sup(u)\f$ demand. |
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41 | A minimum cost flow is an \f$f: A\rightarrow\mathbf{R}\f$ solution |
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42 | of the following optimization problem. |
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43 | |
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44 | \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] |
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45 | \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq |
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46 | sup(u) \quad \forall u\in V \f] |
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47 | \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] |
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48 | |
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49 | The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be |
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50 | zero or negative in order to have a feasible solution (since the sum |
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51 | of the expressions on the left-hand side of the inequalities is zero). |
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52 | It means that the total demand must be greater or equal to the total |
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53 | supply and all the supplies have to be carried out from the supply nodes, |
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54 | but there could be demands that are not satisfied. |
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55 | If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand |
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56 | constraints have to be satisfied with equality, i.e. all demands |
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57 | have to be satisfied and all supplies have to be used. |
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58 | |
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59 | |
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60 | \section mcf_algs Algorithms |
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61 | |
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62 | LEMON contains several algorithms for solving this problem, for more |
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63 | information see \ref min_cost_flow_algs "Minimum Cost Flow Algorithms". |
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64 | |
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65 | A feasible solution for this problem can be found using \ref Circulation. |
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66 | |
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67 | |
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68 | \section mcf_dual Dual Solution |
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69 | |
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70 | The dual solution of the minimum cost flow problem is represented by |
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71 | node potentials \f$\pi: V\rightarrow\mathbf{R}\f$. |
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72 | An \f$f: A\rightarrow\mathbf{R}\f$ primal feasible solution is optimal |
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73 | if and only if for some \f$\pi: V\rightarrow\mathbf{R}\f$ node potentials |
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74 | the following \e complementary \e slackness optimality conditions hold. |
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75 | |
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76 | - For all \f$uv\in A\f$ arcs: |
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77 | - if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$; |
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78 | - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$; |
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79 | - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$. |
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80 | - For all \f$u\in V\f$ nodes: |
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81 | - \f$\pi(u)<=0\f$; |
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82 | - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$, |
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83 | then \f$\pi(u)=0\f$. |
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84 | |
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85 | Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc |
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86 | \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e. |
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87 | \f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f] |
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88 | |
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89 | All algorithms provide dual solution (node potentials), as well, |
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90 | if an optimal flow is found. |
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91 | |
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92 | |
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93 | \section mcf_eq Equality Form |
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94 | |
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95 | The above \ref mcf_def "definition" is actually more general than the |
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96 | usual formulation of the minimum cost flow problem, in which strict |
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97 | equalities are required in the supply/demand contraints. |
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98 | |
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99 | \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] |
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100 | \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) = |
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101 | sup(u) \quad \forall u\in V \f] |
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102 | \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] |
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103 | |
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104 | However if the sum of the supply values is zero, then these two problems |
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105 | are equivalent. |
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106 | The \ref min_cost_flow_algs "algorithms" in LEMON support the general |
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107 | form, so if you need the equality form, you have to ensure this additional |
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108 | contraint manually. |
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109 | |
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110 | |
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111 | \section mcf_leq Opposite Inequalites (LEQ Form) |
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112 | |
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113 | Another possible definition of the minimum cost flow problem is |
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114 | when there are <em>"less or equal"</em> (LEQ) supply/demand constraints, |
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115 | instead of the <em>"greater or equal"</em> (GEQ) constraints. |
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116 | |
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117 | \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] |
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118 | \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \leq |
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119 | sup(u) \quad \forall u\in V \f] |
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120 | \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] |
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121 | |
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122 | It means that the total demand must be less or equal to the |
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123 | total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or |
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124 | positive) and all the demands have to be satisfied, but there |
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125 | could be supplies that are not carried out from the supply |
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126 | nodes. |
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127 | The equality form is also a special case of this form, of course. |
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128 | |
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129 | You could easily transform this case to the \ref mcf_def "GEQ form" |
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130 | of the problem by reversing the direction of the arcs and taking the |
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131 | negative of the supply values (e.g. using \ref ReverseDigraph and |
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132 | \ref NegMap adaptors). |
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133 | However \ref NetworkSimplex algorithm also supports this form directly |
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134 | for the sake of convenience. |
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135 | |
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136 | Note that the optimality conditions for this supply constraint type are |
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137 | slightly differ from the conditions that are discussed for the GEQ form, |
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138 | namely the potentials have to be non-negative instead of non-positive. |
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139 | An \f$f: A\rightarrow\mathbf{R}\f$ feasible solution of this problem |
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140 | is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{R}\f$ |
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141 | node potentials the following conditions hold. |
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142 | |
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143 | - For all \f$uv\in A\f$ arcs: |
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144 | - if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$; |
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145 | - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$; |
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146 | - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$. |
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147 | - For all \f$u\in V\f$ nodes: |
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148 | - \f$\pi(u)>=0\f$; |
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149 | - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$, |
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150 | then \f$\pi(u)=0\f$. |
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151 | |
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152 | */ |
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153 | } |
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