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    pgRouting Manual
    Copyright(c) pgRouting Contributors

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* **Supported versions:**
  `Latest <https://docs.pgrouting.org/latest/en/flow-family.html>`__
  (`3.3 <https://docs.pgrouting.org/3.3/en/flow-family.html>`__)
  `3.2 <https://docs.pgrouting.org/3.2/en/flow-family.html>`__
  `3.1 <https://docs.pgrouting.org/3.1/en/flow-family.html>`__
  `3.0 <https://docs.pgrouting.org/3.0/en/flow-family.html>`__
* **Unsupported versions:**
  `2.6 <https://docs.pgrouting.org/2.6/en/flow-family.html>`__
  `2.5 <https://docs.pgrouting.org/2.5/en/flow-family.html>`__
  `2.4 <https://docs.pgrouting.org/2.4/en/flow-family.html>`__
  `2.3 <https://docs.pgrouting.org/2.3/en/src/max_flow/doc/maxFlow.html>`__

Flow - Family of functions
===================================

.. index from here

* :doc:`pgr_maxFlow` - Only the Max flow calculation using Push and Relabel algorithm.
* :doc:`pgr_boykovKolmogorov` - Boykov and Kolmogorov with details of flow on edges.
* :doc:`pgr_edmondsKarp` - Edmonds and Karp algorithm with details of flow on edges.
* :doc:`pgr_pushRelabel` - Push and relabel algorithm with details of flow on edges.
* Applications

  * :doc:`pgr_edgeDisjointPaths` - Calculates edge disjoint paths between two groups of vertices.
  * :doc:`pgr_maxCardinalityMatch` - Calculates a maximum cardinality matching in a graph.

.. index to here

.. rubric:: Experimental

.. include:: experimental.rst
   :start-after: begin-warn-expr
   :end-before: end-warn-expr

.. index experimental from here

* :doc:`pgr_maxFlowMinCost` - Details of flow and cost on edges.
* :doc:`pgr_maxFlowMinCost_Cost` - Only the Min Cost calculation.

.. index experimental to here

.. toctree::
    :hidden:

    pgr_maxFlow
    pgr_boykovKolmogorov
    pgr_edmondsKarp
    pgr_pushRelabel
    pgr_edgeDisjointPaths
    pgr_maxCardinalityMatch
    pgr_maxFlowMinCost
    pgr_maxFlowMinCost_Cost


Flow Functions General Information
-----------------------------------

.. characteristics_start

**The main characteristics are:**

- The graph is **directed**.
- Process is done only on edges with positive capacities.
- When the maximum flow is 0 then there is no flow and **EMPTY SET** is returned.

  - There is no flow when a **source** is the same as a **target**.

- Any duplicated value in the source(s) or target(s) are ignored.
- Calculates the flow/residual capacity for each edge. In the output

  - Edges with zero flow are omitted.

- Creates a **super source** and edges to all the source(s), and a **super target** and the edges from all the targets(s).
- The maximum flow through the graph is guaranteed to be the value returned by :doc:`pgr_maxFlow <pgr_maxFlow>` when executed with the same parameters and can be calculated:

  - By aggregation of the outgoing flow from the sources
  - By aggregation of the incoming flow to the targets

.. characteristics_end


:doc:`pgr_maxFlow <pgr_maxFlow>`  is the  maximum Flow and that maximum is guaranteed to be the same on the functions :doc:`pgr_pushRelabel <pgr_pushRelabel>`, :doc:`pgr_edmondsKarp <pgr_edmondsKarp>`, :doc:`pgr_boykovKolmogorov <pgr_boykovKolmogorov>`, but the actual flow through each edge may vary.

Parameters
-------------------------------------------------------------------------------

.. pgr_flow_parameters_start

===================== ================== ======== =================================================
Column                Type               Default     Description
===================== ================== ======== =================================================
**Edges SQL**         ``TEXT``                    Edges query as described in `Inner Queries`_.
**Combinations SQL**  ``TEXT``                    Combinations query as described in `Inner Queries`_.
**source**            ``BIGINT``                  Identifier of the starting vertex of the flow.
**sources**           ``ARRAY[BIGINT]``           Array of identifiers of the starting vertices of the flow.
**target**            ``BIGINT``                  Identifier of the ending vertex of the flow.
**targets**           ``ARRAY[BIGINT]``           Array of identifiers of the ending vertices of the flow.
===================== ================== ======== =================================================

.. pgr_flow_parameters_end

Inner queries
-------------------------------------------------------------------------------

.. rubric:: For :doc:`pgr_pushRelabel <pgr_pushRelabel>`, :doc:`pgr_edmondsKarp <pgr_edmondsKarp>`, :doc:`pgr_boykovKolmogorov <pgr_boykovKolmogorov>` :

.. flow_edges_sql_start

:Edges SQL: an SQL query of a directed graph of capacities, which should return a set of rows with the following columns:

==================== =================== ======== =================================================
Column               Type                Default  Description
==================== =================== ======== =================================================
**id**               ``ANY-INTEGER``              Identifier of the edge.
**source**           ``ANY-INTEGER``              Identifier of the first end point vertex of the edge.
**target**           ``ANY-INTEGER``              Identifier of the second end point vertex of the edge.
**capacity**         ``ANY-INTEGER``              Weight of the edge  `(source, target)`

                                                  - When negative: edge `(source, target)` does not exist, therefore it's not part of the graph.

**reverse_capacity** ``ANY-INTEGER``       -1     Weight of the edge `(target, source)`,

                                                  - When negative: edge `(target, source)` does not exist, therefore it's not part of the graph.

==================== =================== ======== =================================================

Where:

:ANY-INTEGER: SMALLINT, INTEGER, BIGINT

.. flow_edges_sql_end

For :doc:`pgr_maxFlowMinCost` and :doc:`pgr_maxFlowMinCost_Cost`:

.. costFlow_edges_sql_start

:Edges SQL: an SQL query of a directed graph of capacities, which should return a set of rows with the following columns:

==================== =================== ======== =================================================
Column               Type                Default  Description
==================== =================== ======== =================================================
**id**               ``ANY-INTEGER``              Identifier of the edge.
**source**           ``ANY-INTEGER``              Identifier of the first end point vertex of the edge.
**target**           ``ANY-INTEGER``              Identifier of the second end point vertex of the edge.
**capacity**         ``ANY-INTEGER``              Capacity of the edge  `(source, target)`

                                                  * When negative: edge `(source, target)` does not exist, therefore it's not part of the graph.
**reverse_capacity** ``ANY-INTEGER``       -1     Capacity of the edge `(target, source)`,

                                                  * When negative: edge `(target, source)` does not exist, therefore it's not part of the graph.
**cost**             ``ANY-NUMERICAL``            Weight of the edge  `(source, target)` if it exists.
**reverse_cost**     ``ANY-NUMERICAL``     0      Weight of the edge `(target, source)` if it exists.
==================== =================== ======== =================================================

Where:

:ANY-INTEGER: SMALLINT, INTEGER, BIGINT
:ANY-NUMERICAL: smallint, int, bigint, real, float

.. costFlow_edges_sql_end

For :doc:`pgr_pushRelabel <pgr_pushRelabel>`, :doc:`pgr_edmondsKarp <pgr_edmondsKarp>`, :doc:`pgr_boykovKolmogorov <pgr_boykovKolmogorov>`, :doc:`pgr_edgeDisjointPaths <pgr_edgeDisjointPaths>`, :doc:`pgr_maxFlowMinCost <pgr_maxFlowMinCost>` and :doc:`pgr_maxFlowMinCost_Cost <pgr_maxFlowMinCost_Cost>` :

.. flow_combinations_sql_start

:Combinations SQL: an SQL query which should return a set of rows with the following columns:

.. include:: pgRouting-concepts.rst
    :start-after: basic_combinations_sql_start
    :end-before: basic_combinations_sql_end

The function aggregates the sources and the targets, removes the duplicates, and then it calculates the result from the resultant source vertices to the target vertices.

.. flow_combinations_sql_end

Result Columns
-------------------------------------------------------------------------------

.. rubric:: For :doc:`pgr_pushRelabel <pgr_pushRelabel>`, :doc:`pgr_edmondsKarp <pgr_edmondsKarp>`, :doc:`pgr_boykovKolmogorov <pgr_boykovKolmogorov>` :

.. result_flow_start

=====================  ====================  =================================================
Column                 Type                  Description
=====================  ====================  =================================================
**seq**                ``INT``               Sequential value starting from **1**.
**edge**               ``BIGINT``            Identifier of the edge in the original query(edges_sql).
**start_vid**          ``BIGINT``            Identifier of the first end point vertex of the edge.
**end_vid**            ``BIGINT``            Identifier of the second end point vertex of the edge.
**flow**               ``BIGINT``            Flow through the edge in the direction (``start_vid``, ``end_vid``).
**residual_capacity**  ``BIGINT``            Residual capacity of the edge in the direction (``start_vid``, ``end_vid``).
=====================  ====================  =================================================

.. result_flow_end

.. rubric:: For :doc:`pgr_maxFlowMinCost`

.. result_costFlow_start

=====================  ====================  =================================================
Column                 Type                  Description
=====================  ====================  =================================================
**seq**                ``INT``               Sequential value starting from **1**.
**edge**               ``BIGINT``            Identifier of the edge in the original query(edges_sql).
**source**             ``BIGINT``            Identifier of the first end point vertex of the edge.
**target**             ``BIGINT``            Identifier of the second end point vertex of the edge.
**flow**               ``BIGINT``            Flow through the edge in the direction (source, target).
**residual_capacity**  ``BIGINT``            Residual capacity of the edge in the direction (source, target).
**cost**               ``FLOAT``             The cost of sending this flow through the edge in the direction (source, target).
**agg_cost**           ``FLOAT``             The aggregate cost.
=====================  ====================  =================================================

.. result_costFlow_end


Adcanced Documentation
------------------------

A flow network is a directed graph where each edge has a capacity and a flow.
The flow through an edge must not exceed the capacity of the edge.
Additionally, the incoming and outgoing flow of a node must be equal except
for source which only has outgoing flow, and the destination(sink) which only has incoming flow.

Maximum flow algorithms calculate the maximum flow through the graph and the flow of each edge.

The maximum flow through the graph is guaranteed to be the same with all implementations,
but the actual flow through each edge may vary.
Given the following query:

pgr_maxFlow :math:`(edges\_sql, source\_vertex, sink\_vertex)`

where :math:`edges\_sql = \{(id_i, source_i, target_i, capacity_i, reverse\_capacity_i)\}`

.. rubric:: Graph definition

The weighted directed graph, :math:`G(V,E)`, is defined as:

* the set of vertices  :math:`V`

  - :math:`source\_vertex  \cup  sink\_vertex  \bigcup  source_i  \bigcup  target_i`

* the set of edges :math:`E`

  - :math:`E = \begin{cases}
    \text{ } \{(source_i, target_i, capacity_i) \text{ when } capacity > 0 \} & \quad  \text{ if } reverse\_capacity = \varnothing \\
    \text{ } & \quad \text{ } \\
    \{(source_i, target_i, capacity_i) \text{ when } capacity > 0 \} & \text{ } \\
    \cup \{(target_i, source_i, reverse\_capacity_i) \text{ when } reverse\_capacity_i > 0)\} & \quad \text{ if } reverse\_capacity \neq \varnothing \\ \end{cases}`


.. rubric:: Maximum flow problem

Given:


- :math:`G(V,E)`
- :math:`source\_vertex \in V` the source vertex
- :math:`sink\_vertex \in V` the sink vertex

Then:

- :math:`pgr\_maxFlow(edges\_sql, source, sink) = \boldsymbol{\Phi}`

- :math:`\boldsymbol{\Phi} = {(id_i, edge\_id_i, source_i, target_i, flow_i, residual\_capacity_i)}`

Where:

:math:`\boldsymbol{\Phi}` is a subset of the original edges with their residual capacity and flow.
The maximum flow through the graph can be obtained by aggregating on the source or sink
and summing the flow from/to it. In particular:

- :math:`id_i = i`
- :math:`edge\_id = id_i` in edges_sql
- :math:`residual\_capacity_i = capacity_i - flow_i`


See Also
--------

* https://en.wikipedia.org/wiki/Maximum_flow_problem

.. rubric:: Indices and tables

* :ref:`genindex`
* :ref:`search`
