pgr_maxFlowMinCost  Experimental¶
pgr_maxFlowMinCost
— Calculates the flow on the graph edges that maximizes
the flow and minimizes the cost from the sources to the targets.
Warning
Possible server crash
These functions might create a server crash
Warning
Experimental functions
They are not officially of the current release.
They likely will not be officially be part of the next release:
The functions might not make use of ANYINTEGER and ANYNUMERICAL
Name might change.
Signature might change.
Functionality might change.
pgTap tests might be missing.
Might need c/c++ coding.
May lack documentation.
Documentation if any might need to be rewritten.
Documentation examples might need to be automatically generated.
Might need a lot of feedback from the comunity.
Might depend on a proposed function of pgRouting
Might depend on a deprecated function of pgRouting
Availability
Version 3.0.0
New experimental function
Support
Description¶
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 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
TODO check which statement is true:
The cost value of all input edges must be nonnegative.
Process is done when the cost value of all input edges is nonnegative.
Process is done on edges with nonnegative cost.
Running time: \(O(U * (E + V * logV))\)
where \(U\) is the value of the max flow.
\(U\) is upper bound on number of iterations. In many real world cases number of iterations is much smaller than \(U\).
Signatures¶
Summary
pgr_maxFlowMinCost(Edges SQL, source, target)
pgr_maxFlowMinCost(Edges SQL, sources, target)
pgr_maxFlowMinCost(Edges SQL, source, targets)
pgr_maxFlowMinCost(Edges SQL, sources, targets)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
One to One¶
pgr_maxFlowMinCost(Edges SQL, source, target)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
 Example
From vertex \(2\) to vertex \(3\)
SELECT * FROM pgr_MaxFlowMinCost(
'SELECT id,
source, target,
capacity, reverse_capacity,
cost, reverse_cost FROM edge_table',
2, 3
);
seq  edge  source  target  flow  residual_capacity  cost  agg_cost
+++++++
1  4  2  5  80  20  80  80
2  3  4  3  80  50  80  160
3  8  5  6  80  20  80  240
4  9  6  9  80  50  80  320
5  16  9  4  80  0  80  400
(5 rows)
One to Many¶
pgr_maxFlowMinCost(Edges SQL, source, targets)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
 Example
From vertex \(13\) to vertices \(\{7, 1, 4\}\)
SELECT * FROM pgr_MaxFlowMinCost(
'SELECT id,
source, target,
capacity, reverse_capacity,
cost, reverse_cost FROM edge_table',
13, ARRAY[7, 1, 4]
);
seq  edge  source  target  flow  residual_capacity  cost  agg_cost
+++++++
1  1  2  1  50  80  50  50
2  4  5  2  50  0  50  100
3  16  9  4  50  30  50  150
4  10  10  5  50  0  50  200
5  12  10  11  50  50  50  250
6  13  11  12  50  50  50  300
7  15  12  9  50  0  50  350
8  14  13  10  100  30  100  450
(8 rows)
Many to One¶
pgr_maxFlowMinCost(Edges SQL, sources, target)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
 Example
From vertices \(\{1, 7, 14\}\) to vertex \(12\)
SELECT * FROM pgr_MaxFlowMinCost(
'SELECT id,
source, target,
capacity, reverse_capacity,
cost, reverse_cost FROM edge_table',
ARRAY[1, 7, 14], 12
);
seq  edge  source  target  flow  residual_capacity  cost  agg_cost
+++++++
1  1  1  2  80  0  80  80
2  4  2  5  80  20  80  160
3  8  5  6  100  0  100  260
4  10  5  10  30  100  30  290
5  9  6  9  50  80  50  340
6  11  6  11  50  80  50  390
7  6  7  8  50  0  50  440
8  7  8  5  50  0  50  490
9  15  9  12  50  30  50  540
10  12  10  11  30  70  30  570
11  13  11  12  80  20  80  650
(11 rows)
Many to Many¶
pgr_maxFlowMinCost(Edges SQL, sources, targets)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
 Example
From vertices \(\{7, 13\}\) to vertices \(\{3, 9\}\)
SELECT * FROM pgr_MaxFlowMinCost(
'SELECT id,
source, target,
capacity, reverse_capacity,
cost, reverse_cost FROM edge_table',
ARRAY[7, 13], ARRAY[3, 9]
);
seq  edge  source  target  flow  residual_capacity  cost  agg_cost
+++++++
1  8  5  6  100  0  100  100
2  9  6  9  100  30  100  200
3  6  7  8  50  0  50  250
4  7  8  5  50  0  50  300
5  10  10  5  50  0  50  350
6  12  10  11  50  50  50  400
7  13  11  12  50  50  50  450
8  15  12  9  50  0  50  500
9  14  13  10  100  30  100  600
(9 rows)
Parameters¶
Column 
Type 
Default 
Description 

Edges SQL 

The edges SQL query as described in Inner Query. 

source 

Identifier of the starting vertex of the flow. 

sources 

Array of identifiers of the starting vertices of the flow. 

target 

Identifier of the ending vertex of the flow. 

targets 

Array of identifiers of the ending vertices of the flow. 
Inner query¶
 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 

Identifier of the edge. 

source 

Identifier of the first end point vertex of the edge. 

target 

Identifier of the second end point vertex of the edge. 

capacity 

Capacity of the edge (source, target)


reverse_capacity 

1 
Capacity of the edge (target, source),

cost 

Weight of the edge (source, target) if it exists. 

reverse_cost 

0 
Weight of the edge (target, source) if it exists. 
Where:
 ANYINTEGER
SMALLINT, INTEGER, BIGINT
 ANYNUMERICAL
smallint, int, bigint, real, float
Result Columns¶
Column 
Type 
Description 

seq 

Sequential value starting from 1. 
edge 

Identifier of the edge in the original query(edges_sql). 
source 

Identifier of the first end point vertex of the edge. 
target 

Identifier of the second end point vertex of the edge. 
flow 

Flow through the edge in the direction (source, target). 
residual_capacity 

Residual capacity of the edge in the direction (source, target). 
cost 

The cost of sending this flow through the edge in the direction (source, target). 
agg_cost 

The aggregate cost. 