doc/min_cost_flow.dox
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     1 /* -*- mode: C++; indent-tabs-mode: nil; -*-

     2  *

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

     4  *

     5  * Copyright (C) 2003-2009

     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 \ref amo93networkflows.

    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 left-hand 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$\pi(u)\leq 0\f$;

    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 non-negative instead of non-positive.

   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$\pi(u)\geq 0\f$;

   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 }