
1 /* * mode: C++; indenttabsmode: nil; * 

2 * 

3 * This file is a part of LEMON, a generic C++ optimization library. 

4 * 

5 * Copyright (C) 20032009 

6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport 

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

8 * 

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

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

11 * precise terms see the accompanying LICENSE file. 

12 * 

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

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

15 * purpose. 

16 * 

17 */ 

18 

19 namespace lemon { 

20 

21 /** 

22 \page min_cost_flow Minimum Cost Flow Problem 

23 

24 \section mcf_def Definition (GEQ form) 

25 

26 The \e minimum \e cost \e flow \e problem is to find a feasible flow of 

27 minimum total cost from a set of supply nodes to a set of demand nodes 

28 in a network with capacity constraints (lower and upper bounds) 

29 and arc costs. 

30 

31 Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{R}\f$, 

32 \f$upper: A\rightarrow\mathbf{R}\cup\{+\infty\}\f$ denote the lower and 

33 upper bounds for the flow values on the arcs, for which 

34 \f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$, 

35 \f$cost: A\rightarrow\mathbf{R}\f$ denotes the cost per unit flow 

36 on the arcs and \f$sup: V\rightarrow\mathbf{R}\f$ denotes the 

37 signed supply values of the nodes. 

38 If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$ 

39 supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with 

40 \f$sup(u)\f$ demand. 

41 A minimum cost flow is an \f$f: A\rightarrow\mathbf{R}\f$ solution 

42 of the following optimization problem. 

43 

44 \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] 

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

46 sup(u) \quad \forall u\in V \f] 

47 \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] 

48 

49 The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be 

50 zero or negative in order to have a feasible solution (since the sum 

51 of the expressions on the lefthand side of the inequalities is zero). 

52 It means that the total demand must be greater or equal to the total 

53 supply and all the supplies have to be carried out from the supply nodes, 

54 but there could be demands that are not satisfied. 

55 If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand 

56 constraints have to be satisfied with equality, i.e. all demands 

57 have to be satisfied and all supplies have to be used. 

58 

59 

60 \section mcf_algs Algorithms 

61 

62 LEMON contains several algorithms for solving this problem, for more 

63 information see \ref min_cost_flow_algs "Minimum Cost Flow Algorithms". 

64 

65 A feasible solution for this problem can be found using \ref Circulation. 

66 

67 

68 \section mcf_dual Dual Solution 

69 

70 The dual solution of the minimum cost flow problem is represented by 

71 node potentials \f$\pi: V\rightarrow\mathbf{R}\f$. 

72 An \f$f: A\rightarrow\mathbf{R}\f$ primal feasible solution is optimal 

73 if and only if for some \f$\pi: V\rightarrow\mathbf{R}\f$ node potentials 

74 the following \e complementary \e slackness optimality conditions hold. 

75 

76  For all \f$uv\in A\f$ arcs: 

77  if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$; 

78  if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$; 

79  if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$. 

80  For all \f$u\in V\f$ nodes: 

81  \f$\pi(u)<=0\f$; 

82  if \f$\sum_{uv\in A} f(uv)  \sum_{vu\in A} f(vu) \neq sup(u)\f$, 

83 then \f$\pi(u)=0\f$. 

84 

85 Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc 

86 \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e. 

87 \f[ cost^\pi(uv) = cost(uv) + \pi(u)  \pi(v).\f] 

88 

89 All algorithms provide dual solution (node potentials), as well, 

90 if an optimal flow is found. 

91 

92 

93 \section mcf_eq Equality Form 

94 

95 The above \ref mcf_def "definition" is actually more general than the 

96 usual formulation of the minimum cost flow problem, in which strict 

97 equalities are required in the supply/demand contraints. 

98 

99 \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] 

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

101 sup(u) \quad \forall u\in V \f] 

102 \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] 

103 

104 However if the sum of the supply values is zero, then these two problems 

105 are equivalent. 

106 The \ref min_cost_flow_algs "algorithms" in LEMON support the general 

107 form, so if you need the equality form, you have to ensure this additional 

108 contraint manually. 

109 

110 

111 \section mcf_leq Opposite Inequalites (LEQ Form) 

112 

113 Another possible definition of the minimum cost flow problem is 

114 when there are <em>"less or equal"</em> (LEQ) supply/demand constraints, 

115 instead of the <em>"greater or equal"</em> (GEQ) constraints. 

116 

117 \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] 

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

119 sup(u) \quad \forall u\in V \f] 

120 \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] 

121 

122 It means that the total demand must be less or equal to the 

123 total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or 

124 positive) and all the demands have to be satisfied, but there 

125 could be supplies that are not carried out from the supply 

126 nodes. 

127 The equality form is also a special case of this form, of course. 

128 

129 You could easily transform this case to the \ref mcf_def "GEQ form" 

130 of the problem by reversing the direction of the arcs and taking the 

131 negative of the supply values (e.g. using \ref ReverseDigraph and 

132 \ref NegMap adaptors). 

133 However \ref NetworkSimplex algorithm also supports this form directly 

134 for the sake of convenience. 

135 

136 Note that the optimality conditions for this supply constraint type are 

137 slightly differ from the conditions that are discussed for the GEQ form, 

138 namely the potentials have to be nonnegative instead of nonpositive. 

139 An \f$f: A\rightarrow\mathbf{R}\f$ feasible solution of this problem 

140 is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{R}\f$ 

141 node potentials the following conditions hold. 

142 

143  For all \f$uv\in A\f$ arcs: 

144  if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$; 

145  if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$; 

146  if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$. 

147  For all \f$u\in V\f$ nodes: 

148  \f$\pi(u)>=0\f$; 

149  if \f$\sum_{uv\in A} f(uv)  \sum_{vu\in A} f(vu) \neq sup(u)\f$, 

150 then \f$\pi(u)=0\f$. 

151 

152 */ 

153 } 