1.1 --- a/doc/min_cost_flow.dox	Fri Aug 09 11:07:27 2013 +0200
     1.2 +++ b/doc/min_cost_flow.dox	Sun Aug 11 15:28:12 2013 +0200
     1.3 @@ -2,7 +2,7 @@
     1.4   *
     1.5   * This file is a part of LEMON, a generic C++ optimization library.
     1.6   *
     1.7 - * Copyright (C) 2003-2009
     1.8 + * Copyright (C) 2003-2010
     1.9   * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
    1.10   * (Egervary Research Group on Combinatorial Optimization, EGRES).
    1.11   *
    1.12 @@ -26,7 +26,7 @@
    1.13  The \e minimum \e cost \e flow \e problem is to find a feasible flow of
    1.14  minimum total cost from a set of supply nodes to a set of demand nodes
    1.15  in a network with capacity constraints (lower and upper bounds)
    1.16 -and arc costs.
    1.17 +and arc costs \ref amo93networkflows.
    1.18  
    1.19  Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{R}\f$,
    1.20  \f$upper: A\rightarrow\mathbf{R}\cup\{+\infty\}\f$ denote the lower and
    1.21 @@ -78,10 +78,10 @@
    1.22     - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$;
    1.23     - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
    1.24   - For all \f$u\in V\f$ nodes:
    1.25 -   - \f$\pi(u)<=0\f$;
    1.26 +   - \f$\pi(u)\leq 0\f$;
    1.27     - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
    1.28       then \f$\pi(u)=0\f$.
    1.29 - 
    1.30 +
    1.31  Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc
    1.32  \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.
    1.33  \f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f]
    1.34 @@ -119,7 +119,7 @@
    1.35      sup(u) \quad \forall u\in V \f]
    1.36  \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
    1.37  
    1.38 -It means that the total demand must be less or equal to the 
    1.39 +It means that the total demand must be less or equal to the
    1.40  total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or
    1.41  positive) and all the demands have to be satisfied, but there
    1.42  could be supplies that are not carried out from the supply
    1.43 @@ -145,7 +145,7 @@
    1.44     - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$;
    1.45     - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
    1.46   - For all \f$u\in V\f$ nodes:
    1.47 -   - \f$\pi(u)>=0\f$;
    1.48 +   - \f$\pi(u)\geq 0\f$;
    1.49     - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
    1.50       then \f$\pi(u)=0\f$.
    1.51