RhodeCode

fastest method for computing a maximum flow. All implementations

also provide functions to query the minimum cut, which is the dual

problem of maximum flow.

\ref Circulation is a preflow push-relabel algorithm implemented directly 

for finding feasible circulations, which is a somewhat different problem,

but it is strongly related to maximum flow.

For more information, see \ref Circulation.

*/

/**

@defgroup min_cost_flow_algs Minimum Cost Flow Algorithms

@ingroup algs

\brief Algorithms for finding minimum cost flows and circulations.

This group contains the algorithms for finding minimum cost flows and

circulations \ref amo93networkflows. For more information about this

problem and its dual solution, see \ref min_cost_flow

"Minimum Cost Flow Problem".

LEMON contains several algorithms for this problem.

 - \ref NetworkSimplex Primal Network Simplex algorithm with various

   pivot strategies \ref dantzig63linearprog, \ref kellyoneill91netsimplex.

   \ref bunnagel98efficient.

In general NetworkSimplex is the most efficient implementation,

but in special cases other algorithms could be faster.

For example, if the total supply and/or capacities are rather small,

CapacityScaling is usually the fastest algorithm (without effective scaling).

*/

/**

@defgroup min_cut Minimum Cut Algorithms

@ingroup algs

\brief Algorithms for finding minimum cut in graphs.

This group contains the algorithms for finding minimum cut in graphs.

The \e minimum \e cut \e problem is to find a non-empty and non-complete

\f$X\f$ subset of the nodes with minimum overall capacity on

outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a

\f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum

cut is the \f$X\f$ solution of the next optimization problem:

\f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}

    \sum_{uv\in A: u\in X, v\not\in X}cap(uv) \f]

    /// The type of the digraph

    typedef GR Digraph;

    /// The type of the flow amounts, capacity bounds and supply values

    typedef V Value;

    /// The type of the arc costs

    typedef C Cost;

    /// \brief The type of the heap used for internal Dijkstra computations.

    ///

    /// The type of the heap used for internal Dijkstra computations.

    /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,

    /// its priority type must be \c Cost and its cross reference type

    /// must be \ref RangeMap "RangeMap<int>".

    typedef BinHeap<Cost, RangeMap<int> > Heap;

  };

  /// \addtogroup min_cost_flow_algs

  /// @{

  /// \brief Implementation of the Capacity Scaling algorithm for

  /// finding a \ref min_cost_flow "minimum cost flow".

  ///

  /// \ref CapacityScaling implements the capacity scaling version

  /// of the successive shortest path algorithm for finding a

  /// solution method.

  ///

  /// Most of the parameters of the problem (except for the digraph)

  /// can be given using separate functions, and the algorithm can be

  /// executed using the \ref run() function. If some parameters are not

  /// specified, then default values will be used.

  ///

  /// \tparam GR The digraph type the algorithm runs on.

  /// \tparam V The number type used for flow amounts, capacity bounds

  /// and supply values in the algorithm. By default it is \c int.

  /// \tparam C The number type used for costs and potentials in the

  /// algorithm. By default it is the same as \c V.

  ///

  /// \warning Both number types must be signed and all input data must

  /// be integer.

  /// \warning This algorithm does not support negative costs for such

  /// arcs that have infinite upper bound.

#ifdef DOXYGEN

  template <typename GR, typename V, typename C, typename TR>

#else

  template < typename GR, typename V = int, typename C = V,

             typename TR = CapacityScalingDefaultTraits<GR, V, C> >

#endif

  class CapacityScaling

  };

  // Default traits class for integer cost types

  template <typename GR, typename V, typename C>

  struct CostScalingDefaultTraits<GR, V, C, true>

  {

    typedef GR Digraph;

    typedef V Value;

    typedef C Cost;

#ifdef LEMON_HAVE_LONG_LONG

    typedef long long LargeCost;

#else

    typedef long LargeCost;

#endif

  };

  /// \addtogroup min_cost_flow_algs

  /// @{

  /// \brief Implementation of the Cost Scaling algorithm for

  /// finding a \ref min_cost_flow "minimum cost flow".

  ///

  /// \ref CostScaling implements a cost scaling algorithm that performs

  /// can be viewed as the generalization of the \ref Preflow

  /// "preflow push-relabel" algorithm for the maximum flow problem.

  ///

  /// Most of the parameters of the problem (except for the digraph)

  /// can be given using separate functions, and the algorithm can be

  /// executed using the \ref run() function. If some parameters are not

  /// specified, then default values will be used.

  ///

  /// \tparam GR The digraph type the algorithm runs on.

  /// \tparam V The number type used for flow amounts, capacity bounds

  /// and supply values in the algorithm. By default it is \c int.

  /// \tparam C The number type used for costs and potentials in the

  /// algorithm. By default it is the same as \c V.

  ///

  /// \warning Both number types must be signed and all input data must

  /// be integer.

  /// \warning This algorithm does not support negative costs for such

  /// arcs that have infinite upper bound.

  ///

  /// \note %CostScaling provides three different internal methods,

  /// from which the most efficient one is used by default.

  /// For more information, see \ref Method.

#ifdef DOXYGEN

  template <typename GR, typename V, typename C, typename TR>