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1 /* -*- mode: C++; indent-tabs-mode: nil; -*- |
1 /* -*- mode: C++; indent-tabs-mode: nil; -*- |
2 * |
2 * |
3 * This file is a part of LEMON, a generic C++ optimization library. |
3 * This file is a part of LEMON, a generic C++ optimization library. |
4 * |
4 * |
5 * Copyright (C) 2003-2009 |
5 * Copyright (C) 2003-2011 |
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
7 * (Egervary Research Group on Combinatorial Optimization, EGRES). |
7 * (Egervary Research Group on Combinatorial Optimization, EGRES). |
8 * |
8 * |
9 * Permission to use, modify and distribute this software is granted |
9 * Permission to use, modify and distribute this software is granted |
10 * provided that this copyright notice appears in all copies. For |
10 * provided that this copyright notice appears in all copies. For |
79 - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$. |
79 - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$. |
80 - For all \f$u\in V\f$ nodes: |
80 - For all \f$u\in V\f$ nodes: |
81 - \f$\pi(u)<=0\f$; |
81 - \f$\pi(u)<=0\f$; |
82 - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$, |
82 - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$, |
83 then \f$\pi(u)=0\f$. |
83 then \f$\pi(u)=0\f$. |
84 |
84 |
85 Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc |
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. |
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] |
87 \f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f] |
88 |
88 |
89 All algorithms provide dual solution (node potentials), as well, |
89 All algorithms provide dual solution (node potentials), as well, |
117 \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] |
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 |
118 \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \leq |
119 sup(u) \quad \forall u\in V \f] |
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] |
120 \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] |
121 |
121 |
122 It means that the total demand must be less or equal to the |
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 |
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 |
124 positive) and all the demands have to be satisfied, but there |
125 could be supplies that are not carried out from the supply |
125 could be supplies that are not carried out from the supply |
126 nodes. |
126 nodes. |
127 The equality form is also a special case of this form, of course. |
127 The equality form is also a special case of this form, of course. |