max_flow

The maximum flow problem consists of finding a flow through a graph such that it is the maximum possible flow.

The algorithm implementation is based on the Ford-Fulkerson method with capacity scaling. Ford-Fulkerson method is not itself an algorithm as it does not specify the procedure of finding augmenting paths in a residual graph. It is a greedy method, using augmenting paths as it comes across them. Input is a weighted graph with a defined source and sink, representing the beginning and end of the flow network. The algorithm begins with an empty flow and, at each step, finds a path, called an augmenting path, from the source to the sink that generates more flow. When flow cannot be increased anymore, the algorithm stops, and the maximum flow has been found.

The capacity scaling is a heuristic for finding augmenting paths in such a way that prioritizes taking edges with larger capacities, maintaining a threshold value that is only lowered once no larger path can be found. It speeds up the algorithm noticeably compared to a standard DFS search.

The algorithm is adapted to work with heterogeneous graphs, meaning not all edges need to have the defined edge property used for edge flow. When an edge doesn't have a flow, it is skipped, and when no edges have this property, the returning max flow value is 0.

Trait	Value
Module type	algorithm
Implementation	Python
Graph direction	directed
Edge weights	weighted
Parallelism	sequential

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Procedures

info

If you want to execute this algorithm on graph projections, subgraphs or portions of the graph, be sure to check out the guide on How to run a MAGE module on subgraphs.

get_flow(parameters, edge_property)

Input:

• start_v: Vertex ➡ Source node from which the maximum flow is calculated
• end_v: Vertex ➡ Sink node to which the max flow is calculated
• edge_property: string (default="weight") ➡ Edge property which is used as the flow capacity of the edge

Output:

• max_flow: Number ➡ Maximum flow of the network, from source to sink

Usage:

MATCH (source {id: 0}), (sink {id: 5})
CALL max_flow.get_flow(source, sink, "weight")
YIELD max_flow
RETURN max_flow;

get_paths(parameters, edge_property)

Input:

• start_v: Vertex ➡ Source node from which the maximum flow is calculated
• end_v: Vertex ➡ Sink node to which the max flow is calculated
• edge_property: string (default="weight") ➡ Edge property which is used as the flow capacity of the edge

Output:

• path: Path ➡ path with a flow in a maximum flow
• flow: Number ➡ flow amount corresponding to that path

Usage:

MATCH (source {id: 0}), (sink {id: 5})
CALL max_flow.get_paths(source, sink, "weight")
YIELD path, flow
RETURN path, flow;

Example

Step 1: Input graph

Step 2: Cypher load commands

MERGE (a:Node {id: "A"}) MERGE (b:Node {id: "B"}) CREATE (a)-[:RELATION {weight: 9}]->(b);
MERGE (a:Node {id: "A"}) MERGE (b:Node {id: "C"}) CREATE (a)-[:RELATION {weight: 10}]->(b);
MERGE (a:Node {id: "B"}) MERGE (b:Node {id: "E"}) CREATE (a)-[:RELATION {weight: 8}]->(b);
MERGE (a:Node {id: "C"}) MERGE (b:Node {id: "F"}) CREATE (a)-[:RELATION {weight: 7}]->(b);
MERGE (a:Node {id: "C"}) MERGE (b:Node {id: "D"}) CREATE (a)-[:RELATION {weight: 1}]->(b);
MERGE (a:Node {id: "A"}) MERGE (b:Node {id: "D"}) CREATE (a)-[:RELATION {weight: 8}]->(b);
MERGE (a:Node {id: "E"}) MERGE (b:Node {id: "D"}) CREATE (a)-[:RELATION {weight: 2}]->(b);
MERGE (a:Node {id: "D"}) MERGE (b:Node {id: "F"}) CREATE (a)-[:RELATION {weight: 11}]->(b);
MERGE (a:Node {id: "E"}) MERGE (b:Node {id: "G"}) CREATE (a)-[:RELATION {weight: 5}]->(b);
MERGE (a:Node {id: "F"}) MERGE (b:Node {id: "G"}) CREATE (a)-[:RELATION {weight: 14}]->(b);

Step 3: Running command

MATCH (source {id: "A"}), (sink {id: "G"})
CALL max_flow.get_flow(source, sink)
YIELD max_flow
RETURN max_flow;

Step 4: Results

+----------+
| max_flow |
+----------+
| 19       |
+----------+