CostScaling implements a cost scaling algorithm that performs push/augment and relabel operations for finding a minimum cost flow [1], [18], [19], [2]. It is a highly efficient primaldual solution method, which can be viewed as the generalization of the preflow pushrelabel algorithm for the maximum flow problem. It is a polynomial algorithm, its running time complexity is , where K denotes the maximum arc cost.
In general, NetworkSimplex and CostScaling are the fastest implementations available in LEMON for solving this problem. (For more information, see the module page.)
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 . 
TR  The traits class that defines various types used by the algorithm. By default, it is CostScalingDefaultTraits<GR, V, C>. In most cases, this parameter should not be set directly, consider to use the named template parameters instead. 
V
and C
must be signed number types. #include <lemon/cost_scaling.h>
Classes  
struct  SetLargeCost 
Named parameter for setting LargeCost type. More...  
Public Types  
enum  ProblemType { INFEASIBLE, OPTIMAL, UNBOUNDED } 
Problem type constants for the run() function. More...  
enum  Method { PUSH, AUGMENT, PARTIAL_AUGMENT } 
Constants for selecting the internal method. More...  
typedef TR::Digraph  Digraph 
The type of the digraph.  
typedef TR::Value  Value 
The type of the flow amounts, capacity bounds and supply values.  
typedef TR::Cost  Cost 
The type of the arc costs.  
typedef TR::LargeCost  LargeCost 
The large cost type.  
typedef TR  Traits 
The traits class of the algorithm.  
Public Member Functions  
CostScaling (const GR &graph)  
Constructor.  
Parameters  
The parameters of the algorithm can be specified using these functions.  
template<typename LowerMap >  
CostScaling &  lowerMap (const LowerMap &map) 
Set the lower bounds on the arcs.  
template<typename UpperMap >  
CostScaling &  upperMap (const UpperMap &map) 
Set the upper bounds (capacities) on the arcs.  
template<typename CostMap >  
CostScaling &  costMap (const CostMap &map) 
Set the costs of the arcs.  
template<typename SupplyMap >  
CostScaling &  supplyMap (const SupplyMap &map) 
Set the supply values of the nodes.  
CostScaling &  stSupply (const Node &s, const Node &t, Value k) 
Set single source and target nodes and a supply value.  
Execution control  
The algorithm can be executed using run().  
ProblemType  run (Method method=PARTIAL_AUGMENT, int factor=16) 
Run the algorithm.  
CostScaling &  resetParams () 
Reset all the parameters that have been given before.  
CostScaling &  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).  
Private Member Functions  
void  startAugment (int max_length) 
Execute the algorithm performing augment and relabel operations.  
void  startPush () 
Execute the algorithm performing push and relabel operations.  
typedef TR::LargeCost LargeCost 
The large cost type used for internal computations. By default, it is long
long
if the Cost
type is integer, otherwise it is double
.
enum ProblemType 
Enum type containing the problem type constants that can be returned by the run() function of the algorithm.
enum Method 
Enum type containing constants for selecting the internal method for the run() function.
CostScaling provides three internal methods that differ mainly in their base operations, which are used in conjunction with the relabel operation. By default, the so called Partial AugmentRelabel method is used, which turned out to be the most efficient and the most robust on various test inputs. However, the other methods can be selected using the run() function with the proper parameter.

inline 
The constructor of the class.
graph  The digraph the algorithm runs on. 

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 runs the algorithm. The paramters can be specified using functions lowerMap(), upperMap(), costMap(), supplyMap(), stSupply(). 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.
method  The internal method that will be used in the algorithm. For more information, see Method. 
factor  The cost scaling factor. It must be at least two. 
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 digraph contains an arc of negative cost and infinite upper bound. It means that the objective function is unbounded on that arc, however, note that it could actually be bounded over the feasible flows, but this algroithm cannot handle these cases.

inline 
This function resets all the paramaters that have been given before using functions lowerMap(), upperMap(), costMap(), supplyMap(), stSupply().
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().
It is useful for multiple run() calls. By default, 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.