NetworkSimplex implements the primal Network Simplex algorithm for finding a minimum cost flow [1], [8], [23]. This algorithm is a highly efficient specialized version of the linear programming simplex method directly for the minimum cost flow problem.
In general, NetworkSimplex and CostScaling are the fastest implementations available in LEMON for solving this problem. (For more information, see the module page.) Furthermore, this class supports both directions of the supply/demand inequality constraints. For more information, see SupplyType.
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 run() function. If some parameters are not specified, then default values will be used.
GR  The digraph type the algorithm runs on. 
V  The number type used for flow amounts, capacity bounds and supply values in the algorithm. By default, it is int . 
C  The number type used for costs and potentials in the algorithm. By default, it is the same as V . 
V
and C
must be signed number types. #include <lemon/network_simplex.h>
Public Types  
enum  ProblemType { INFEASIBLE, OPTIMAL, UNBOUNDED } 
Problem type constants for the run() function. More...  
enum  SupplyType { GEQ, LEQ } 
Constants for selecting the type of the supply constraints. More...  
enum  PivotRule { FIRST_ELIGIBLE, BEST_ELIGIBLE, BLOCK_SEARCH, CANDIDATE_LIST, ALTERING_LIST } 
Constants for selecting the pivot rule. More...  
typedef V  Value 
The type of the flow amounts, capacity bounds and supply values.  
typedef C  Cost 
The type of the arc costs.  
Public Member Functions  
NetworkSimplex (const GR &graph, bool arc_mixing=true)  
Constructor.  
Parameters  
The parameters of the algorithm can be specified using these functions.  
template<typename LowerMap >  
NetworkSimplex &  lowerMap (const LowerMap &map) 
Set the lower bounds on the arcs.  
template<typename UpperMap >  
NetworkSimplex &  upperMap (const UpperMap &map) 
Set the upper bounds (capacities) on the arcs.  
template<typename CostMap >  
NetworkSimplex &  costMap (const CostMap &map) 
Set the costs of the arcs.  
template<typename SupplyMap >  
NetworkSimplex &  supplyMap (const SupplyMap &map) 
Set the supply values of the nodes.  
NetworkSimplex &  stSupply (const Node &s, const Node &t, Value k) 
Set single source and target nodes and a supply value.  
NetworkSimplex &  supplyType (SupplyType supply_type) 
Set the type of the supply constraints.  
Execution Control  
The algorithm can be executed using run().  
ProblemType  run (PivotRule pivot_rule=BLOCK_SEARCH) 
Run the algorithm.  
NetworkSimplex &  resetParams () 
Reset all the parameters that have been given before.  
NetworkSimplex &  reset () 
Reset the internal data structures and all the parameters that have been given before.  
Query Functions  
The results of the algorithm can be obtained using these functions.  
template<typename Number >  
Number  totalCost () const 
Return the total cost of the found flow.  
Value  flow (const Arc &a) const 
Return the flow on the given arc.  
template<typename FlowMap >  
void  flowMap (FlowMap &map) const 
Copy the flow values (the primal solution) into the given map.  
Cost  potential (const Node &n) const 
Return the potential (dual value) of the given node.  
template<typename PotentialMap >  
void  potentialMap (PotentialMap &map) const 
Copy the potential values (the dual solution) into the given map.  
Public Attributes  
const Value  INF 
Constant for infinite upper bounds (capacities).  
enum ProblemType 
Enum type containing the problem type constants that can be returned by the run() function of the algorithm.
enum SupplyType 
Enum type containing constants for selecting the supply type, i.e. the direction of the inequalities in the supply/demand constraints of the minimum cost flow problem.
The default supply type is GEQ
, the LEQ
type can be selected using supplyType(). The equality form is a special case of both supply types.
enum PivotRule 
Enum type containing constants for selecting the pivot rule for the run() function.
NetworkSimplex provides five different implementations for the pivot strategy that significantly affects the running time of the algorithm. According to experimental tests conducted on various problem instances, Block Search and Altering Candidate List rules turned out to be the most efficient. Since Block Search is a simpler strategy that seemed to be slightly more robust, it is used by default. However, another pivot rule can easily be selected using the run() function with the proper parameter.

inline 
The constructor of the class.
graph  The digraph the algorithm runs on. 
arc_mixing  Indicate if the arcs will be stored in a mixed order in the internal data structure. In general, it leads to similar performance as using the original arc order, but it makes the algorithm more robust and in special cases, even significantly faster. Therefore, it is enabled by default. 

inline 
This function sets the lower bounds on the arcs. If it is not used before calling run(), the lower bounds will be set to zero on all arcs.
map  An arc map storing the lower bounds. Its Value type must be convertible to the Value type of the algorithm. 
(*this)

inline 
This function sets the upper bounds (capacities) on the arcs. If it is not used before calling run(), the upper bounds will be set to INF on all arcs (i.e. the flow value will be unbounded from above).
map  An arc map storing the upper bounds. Its Value type must be convertible to the Value type of the algorithm. 
(*this)

inline 
This function sets the costs of the arcs. If it is not used before calling run(), the costs will be set to 1
on all arcs.
map  An arc map storing the costs. Its Value type must be convertible to the Cost type of the algorithm. 
(*this)

inline 
This function sets the supply values of the nodes. If neither this function nor stSupply() is used before calling run(), the supply of each node will be set to zero.
map  A node map storing the supply values. Its Value type must be convertible to the Value type of the algorithm. 
(*this)

inline 
This function sets a single source node and a single target node and the required flow value. If neither this function nor supplyMap() is used before calling run(), the supply of each node will be set to zero.
Using this function has the same effect as using supplyMap() with a map in which k
is assigned to s
, k
is assigned to t
and all other nodes have zero supply value.
s  The source node. 
t  The target node. 
k  The required amount of flow from node s to node t (i.e. the supply of s and the demand of t ). 
(*this)

inline 
This function sets the type of the supply/demand constraints. If it is not used before calling run(), the GEQ supply type will be used.
For more information, see SupplyType.
(*this)

inline 
This function runs the algorithm. The paramters can be specified using functions lowerMap(), upperMap(), costMap(), supplyMap(), stSupply(), supplyType(). For example,
This function can be called more than once. All the given parameters are kept for the next call, unless resetParams() or reset() is used, thus only the modified parameters have to be set again. If the underlying digraph was also modified after the construction of the class (or the last reset() call), then the reset() function must be called.
pivot_rule  The pivot rule that will be used during the algorithm. For more information, see PivotRule. 
INFEASIBLE
if no feasible flow exists, OPTIMAL
if the problem has optimal solution (i.e. it is feasible and bounded), and the algorithm has found optimal flow and node potentials (primal and dual solutions), UNBOUNDED
if the objective function of the problem is unbounded, i.e. there is a directed cycle having negative total cost and infinite upper bound.

inline 
This function resets all the paramaters that have been given before using functions lowerMap(), upperMap(), costMap(), supplyMap(), stSupply(), supplyType().
It is useful for multiple run() calls. Basically, all the given parameters are kept for the next run() call, unless resetParams() or reset() is used. If the underlying digraph was also modified after the construction of the class or the last reset() call, then the reset() function must be used, otherwise resetParams() is sufficient.
For example,
(*this)

inline 
This function resets the internal data structures and all the paramaters that have been given before using functions lowerMap(), upperMap(), costMap(), supplyMap(), stSupply(), supplyType().
It is useful for multiple run() calls. Basically, all the given parameters are kept for the next run() call, unless resetParams() or reset() is used. If the underlying digraph was also modified after the construction of the class or the last reset() call, then the reset() function must be used, otherwise resetParams() is sufficient.
See resetParams() for examples.
(*this)

inline 
This function returns the total cost of the found flow. Its complexity is O(e).
Cost
type of the algorithm, which is the default return type of the function.

inline 
This function returns the flow on the given arc.

inline 
This function copies the flow value on each arc into the given map. The Value
type of the algorithm must be convertible to the Value
type of the map.

inline 
This function returns the potential (dual value) of the given node.

inline 
This function copies the potential (dual value) of each node into the given map. The Cost
type of the algorithm must be convertible to the Value
type of the map.
const Value INF 
Constant for infinite upper bounds (capacities). It is std::numeric_limits<Value>::infinity()
if available, std::numeric_limits<Value>::max()
otherwise.