template<typename GR, typename LM, typename UM, typename SM, typename TR>
class lemon::Circulation< GR, LM, UM, SM, TR >
This class implements a push-relabel algorithm for the network circulation problem. It is to find a feasible circulation when lower and upper bounds are given for the flow values on the arcs and lower bounds are given for the difference between the outgoing and incoming flow at the nodes.
The exact formulation of this problem is the following. Let
be a digraph,
denote the lower and upper bounds on the arcs, for which
holds for all
, and
denotes the signed supply values of the nodes. If
, then
is a supply node with
supply, if
, then
is a demand node with
demand. A feasible circulation is an
solution of the following problem.
The sum of the supply values, i.e.
must be zero or negative in order to have a feasible solution (since the sum of the expressions on the left-hand side of the inequalities is zero). It means that the total demand must be greater or equal to the total supply and all the supplies have to be carried out from the supply nodes, but there could be demands that are not satisfied. If
is zero, then all the supply/demand constraints have to be satisfied with equality, i.e. all demands have to be satisfied and all supplies have to be used.
If you need the opposite inequalities in the supply/demand constraints (i.e. the total demand is less than the total supply and all the demands have to be satisfied while there could be supplies that are not used), then you could easily transform the problem to the above form by reversing the direction of the arcs and taking the negative of the supply values (e.g. using ReverseDigraph and NegMap adaptors).
This algorithm either calculates a feasible circulation, or provides a barrier, which prooves that a feasible soultion cannot exist.
Note that this algorithm also provides a feasible solution for the minimum cost flow problem.
- Template Parameters
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GR | The type of the digraph the algorithm runs on. |
LM | The type of the lower bound map. The default map type is GR::ArcMap<int>. |
UM | The type of the upper bound (capacity) map. The default map type is LM . |
SM | The type of the supply map. The default map type is GR::NodeMap<UM::Value>. |
TR | The traits class that defines various types used by the algorithm. By default, it is CirculationDefaultTraits<GR, LM, UM, SM>. In most cases, this parameter should not be set directly, consider to use the named template parameters instead. |
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| Circulation (const Digraph &graph, const LowerMap &lower, const UpperMap &upper, const SupplyMap &supply) |
| Constructor.
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| ~Circulation () |
| Destructor.
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Circulation & | lowerMap (const LowerMap &map) |
| Sets the lower bound map.
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Circulation & | upperMap (const UpperMap &map) |
| Sets the upper bound (capacity) map.
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Circulation & | supplyMap (const SupplyMap &map) |
| Sets the supply map.
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Circulation & | flowMap (FlowMap &map) |
| Sets the flow map.
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Circulation & | elevator (Elevator &elevator) |
| Sets the elevator used by algorithm.
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const Elevator & | elevator () const |
| Returns a const reference to the elevator.
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Circulation & | tolerance (const Tolerance &tolerance) |
| Sets the tolerance used by the algorithm.
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const Tolerance & | tolerance () const |
| Returns a const reference to the tolerance.
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The simplest way to execute the algorithm is to call run().
If you need better control on the initial solution or the execution, you have to call one of the init() functions first, then the start() function.
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void | init () |
| Initializes the internal data structures.
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void | greedyInit () |
| Initializes the internal data structures using a greedy approach.
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bool | start () |
| Executes the algorithm.
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bool | run () |
| Runs the algorithm.
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The results of the circulation algorithm can be obtained using these functions.
Either run() or start() should be called before using them.
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Value | flow (const Arc &arc) const |
| Returns the flow value on the given arc.
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const FlowMap & | flowMap () const |
| Returns a const reference to the flow map.
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bool | barrier (const Node &node) const |
| Returns true if the given node is in a barrier.
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template<class BarrierMap > |
void | barrierMap (BarrierMap &bar) const |
| Gives back a barrier.
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The feasibility of the results can be checked using these functions.
Either run() or start() should be called before using them.
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bool | checkFlow () const |
| Check if the found flow is a feasible circulation.
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bool | checkBarrier () const |
| Check whether or not the last execution provides a barrier.
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