Changes in doc/min_cost_flow.dox [710:8b0df68370a4:1164:f63ba40a60f4] in lemon

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• doc/min_cost_flow.dox (modified) (6 diffs)

doc/min_cost_flow.dox

@@ -3 +3 @@
  * This file is a part of LEMON, a generic C++ optimization library.
  *
- * Copyright (C) 2003-2009
+ * Copyright (C) 2003-2010
  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
  * (Egervary Research Group on Combinatorial Optimization, EGRES).
@@ -27 +27 @@
 minimum total cost from a set of supply nodes to a set of demand nodes
 in a network with capacity constraints (lower and upper bounds)
-and arc costs.
+and arc costs \ref amo93networkflows.
 
 Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{R}\f$,
@@ -79 +79 @@
    - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
  - For all \f$u\in V\f$ nodes:
-   - \f$\pi(u)<=0\f$;
+   - \f$\pi(u)\leq 0\f$;
    - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
      then \f$\pi(u)=0\f$.
- 
+
 Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc
 \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.
@@ -102 +102 @@
 \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
 
-However if the sum of the supply values is zero, then these two problems
+However, if the sum of the supply values is zero, then these two problems
 are equivalent.
 The \ref min_cost_flow_algs "algorithms" in LEMON support the general
@@ -120 +120 @@
 \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
 
-It means that the total demand must be less or equal to the 
+It means that the total demand must be less or equal to the
 total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or
 positive) and all the demands have to be satisfied, but there
@@ -146 +146 @@
    - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
  - For all \f$u\in V\f$ nodes:
-   - \f$\pi(u)>=0\f$;
+   - \f$\pi(u)\geq 0\f$;
    - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
      then \f$\pi(u)=0\f$.