COIN-OR::LEMON - Graph Library

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  • doc/min_cost_flow.dox

    r710 r956  
    33 * This file is a part of LEMON, a generic C++ optimization library.
    44 *
    5  * Copyright (C) 2003-2009
     5 * Copyright (C) 2003-2010
    66 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
    77 * (Egervary Research Group on Combinatorial Optimization, EGRES).
     
    2727minimum total cost from a set of supply nodes to a set of demand nodes
    2828in a network with capacity constraints (lower and upper bounds)
    29 and arc costs.
     29and arc costs \ref amo93networkflows.
    3030
    3131Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{R}\f$,
     
    7979   - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
    8080 - For all \f$u\in V\f$ nodes:
    81    - \f$\pi(u)<=0\f$;
     81   - \f$\pi(u)\leq 0\f$;
    8282   - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
    8383     then \f$\pi(u)=0\f$.
    84  
     84
    8585Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc
    8686\f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.
     
    120120\f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
    121121
    122 It means that the total demand must be less or equal to the 
     122It means that the total demand must be less or equal to the
    123123total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or
    124124positive) and all the demands have to be satisfied, but there
     
    146146   - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
    147147 - For all \f$u\in V\f$ nodes:
    148    - \f$\pi(u)>=0\f$;
     148   - \f$\pi(u)\geq 0\f$;
    149149   - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
    150150     then \f$\pi(u)=0\f$.
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