"""
Maximum flow (and minimum cut) algorithms on capacitated graphs.
"""
import networkx as nx
from .boykovkolmogorov import boykov_kolmogorov
from .dinitz_alg import dinitz
from .edmondskarp import edmonds_karp
from .preflowpush import preflow_push
from .shortestaugmentingpath import shortest_augmenting_path
from .utils import build_flow_dict
# Define the default flow function for computing maximum flow.
default_flow_func = preflow_push
# Functions that don't support cutoff for minimum cut computations.
flow_funcs = [
boykov_kolmogorov,
dinitz,
edmonds_karp,
preflow_push,
shortest_augmenting_path,
]
__all__ = ["maximum_flow", "maximum_flow_value", "minimum_cut", "minimum_cut_value"]
[docs]def maximum_flow(flowG, _s, _t, capacity="capacity", flow_func=None, **kwargs):
"""Find a maximum single-commodity flow.
Parameters
----------
flowG : NetworkX graph
Edges of the graph are expected to have an attribute called
'capacity'. If this attribute is not present, the edge is
considered to have infinite capacity.
_s : node
Source node for the flow.
_t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: 'capacity'.
flow_func : function
A function for computing the maximum flow among a pair of nodes
in a capacitated graph. The function has to accept at least three
parameters: a Graph or Digraph, a source node, and a target node.
And return a residual network that follows NetworkX conventions
(see Notes). If flow_func is None, the default maximum
flow function (:meth:`preflow_push`) is used. See below for
alternative algorithms. The choice of the default function may change
from version to version and should not be relied on. Default value:
None.
kwargs : Any other keyword parameter is passed to the function that
computes the maximum flow.
Returns
-------
flow_value : integer, float
Value of the maximum flow, i.e., net outflow from the source.
flow_dict : dict
A dictionary containing the value of the flow that went through
each edge.
Raises
------
NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If
the input graph is an instance of one of these two classes, a
NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a
feasible flow on the graph is unbounded above and the function
raises a NetworkXUnbounded.
See also
--------
:meth:`maximum_flow_value`
:meth:`minimum_cut`
:meth:`minimum_cut_value`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
Notes
-----
The function used in the flow_func parameter has to return a residual
network that follows NetworkX conventions:
The residual network :samp:`R` from an input graph :samp:`G` has the
same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
in :samp:`G`.
For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
in :samp:`G` or zero otherwise. If the capacity is infinite,
:samp:`R[u][v]['capacity']` will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in
:samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
:samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.
The flow value, defined as the total flow into :samp:`t`, the sink, is
stored in :samp:`R.graph['flow_value']`. Reachability to :samp:`t` using
only edges :samp:`(u, v)` such that
:samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
:samp:`s`-:samp:`t` cut.
Specific algorithms may store extra data in :samp:`R`.
The function should supports an optional boolean parameter value_only. When
True, it can optionally terminate the algorithm as soon as the maximum flow
value and the minimum cut can be determined.
Examples
--------
>>> G = nx.DiGraph()
>>> G.add_edge("x", "a", capacity=3.0)
>>> G.add_edge("x", "b", capacity=1.0)
>>> G.add_edge("a", "c", capacity=3.0)
>>> G.add_edge("b", "c", capacity=5.0)
>>> G.add_edge("b", "d", capacity=4.0)
>>> G.add_edge("d", "e", capacity=2.0)
>>> G.add_edge("c", "y", capacity=2.0)
>>> G.add_edge("e", "y", capacity=3.0)
maximum_flow returns both the value of the maximum flow and a
dictionary with all flows.
>>> flow_value, flow_dict = nx.maximum_flow(G, "x", "y")
>>> flow_value
3.0
>>> print(flow_dict["x"]["b"])
1.0
You can also use alternative algorithms for computing the
maximum flow by using the flow_func parameter.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> flow_value == nx.maximum_flow(G, "x", "y", flow_func=shortest_augmenting_path)[
... 0
... ]
True
"""
if flow_func is None:
if kwargs:
raise nx.NetworkXError(
"You have to explicitly set a flow_func if"
" you need to pass parameters via kwargs."
)
flow_func = default_flow_func
if not callable(flow_func):
raise nx.NetworkXError("flow_func has to be callable.")
R = flow_func(flowG, _s, _t, capacity=capacity, value_only=False, **kwargs)
flow_dict = build_flow_dict(flowG, R)
return (R.graph["flow_value"], flow_dict)
[docs]def maximum_flow_value(flowG, _s, _t, capacity="capacity", flow_func=None, **kwargs):
"""Find the value of maximum single-commodity flow.
Parameters
----------
flowG : NetworkX graph
Edges of the graph are expected to have an attribute called
'capacity'. If this attribute is not present, the edge is
considered to have infinite capacity.
_s : node
Source node for the flow.
_t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: 'capacity'.
flow_func : function
A function for computing the maximum flow among a pair of nodes
in a capacitated graph. The function has to accept at least three
parameters: a Graph or Digraph, a source node, and a target node.
And return a residual network that follows NetworkX conventions
(see Notes). If flow_func is None, the default maximum
flow function (:meth:`preflow_push`) is used. See below for
alternative algorithms. The choice of the default function may change
from version to version and should not be relied on. Default value:
None.
kwargs : Any other keyword parameter is passed to the function that
computes the maximum flow.
Returns
-------
flow_value : integer, float
Value of the maximum flow, i.e., net outflow from the source.
Raises
------
NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If
the input graph is an instance of one of these two classes, a
NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a
feasible flow on the graph is unbounded above and the function
raises a NetworkXUnbounded.
See also
--------
:meth:`maximum_flow`
:meth:`minimum_cut`
:meth:`minimum_cut_value`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
Notes
-----
The function used in the flow_func parameter has to return a residual
network that follows NetworkX conventions:
The residual network :samp:`R` from an input graph :samp:`G` has the
same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
in :samp:`G`.
For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
in :samp:`G` or zero otherwise. If the capacity is infinite,
:samp:`R[u][v]['capacity']` will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in
:samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
:samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.
The flow value, defined as the total flow into :samp:`t`, the sink, is
stored in :samp:`R.graph['flow_value']`. Reachability to :samp:`t` using
only edges :samp:`(u, v)` such that
:samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
:samp:`s`-:samp:`t` cut.
Specific algorithms may store extra data in :samp:`R`.
The function should supports an optional boolean parameter value_only. When
True, it can optionally terminate the algorithm as soon as the maximum flow
value and the minimum cut can be determined.
Examples
--------
>>> G = nx.DiGraph()
>>> G.add_edge("x", "a", capacity=3.0)
>>> G.add_edge("x", "b", capacity=1.0)
>>> G.add_edge("a", "c", capacity=3.0)
>>> G.add_edge("b", "c", capacity=5.0)
>>> G.add_edge("b", "d", capacity=4.0)
>>> G.add_edge("d", "e", capacity=2.0)
>>> G.add_edge("c", "y", capacity=2.0)
>>> G.add_edge("e", "y", capacity=3.0)
maximum_flow_value computes only the value of the
maximum flow:
>>> flow_value = nx.maximum_flow_value(G, "x", "y")
>>> flow_value
3.0
You can also use alternative algorithms for computing the
maximum flow by using the flow_func parameter.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> flow_value == nx.maximum_flow_value(
... G, "x", "y", flow_func=shortest_augmenting_path
... )
True
"""
if flow_func is None:
if kwargs:
raise nx.NetworkXError(
"You have to explicitly set a flow_func if"
" you need to pass parameters via kwargs."
)
flow_func = default_flow_func
if not callable(flow_func):
raise nx.NetworkXError("flow_func has to be callable.")
R = flow_func(flowG, _s, _t, capacity=capacity, value_only=True, **kwargs)
return R.graph["flow_value"]
[docs]def minimum_cut(flowG, _s, _t, capacity="capacity", flow_func=None, **kwargs):
"""Compute the value and the node partition of a minimum (s, t)-cut.
Use the max-flow min-cut theorem, i.e., the capacity of a minimum
capacity cut is equal to the flow value of a maximum flow.
Parameters
----------
flowG : NetworkX graph
Edges of the graph are expected to have an attribute called
'capacity'. If this attribute is not present, the edge is
considered to have infinite capacity.
_s : node
Source node for the flow.
_t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: 'capacity'.
flow_func : function
A function for computing the maximum flow among a pair of nodes
in a capacitated graph. The function has to accept at least three
parameters: a Graph or Digraph, a source node, and a target node.
And return a residual network that follows NetworkX conventions
(see Notes). If flow_func is None, the default maximum
flow function (:meth:`preflow_push`) is used. See below for
alternative algorithms. The choice of the default function may change
from version to version and should not be relied on. Default value:
None.
kwargs : Any other keyword parameter is passed to the function that
computes the maximum flow.
Returns
-------
cut_value : integer, float
Value of the minimum cut.
partition : pair of node sets
A partitioning of the nodes that defines a minimum cut.
Raises
------
NetworkXUnbounded
If the graph has a path of infinite capacity, all cuts have
infinite capacity and the function raises a NetworkXError.
See also
--------
:meth:`maximum_flow`
:meth:`maximum_flow_value`
:meth:`minimum_cut_value`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
Notes
-----
The function used in the flow_func parameter has to return a residual
network that follows NetworkX conventions:
The residual network :samp:`R` from an input graph :samp:`G` has the
same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
in :samp:`G`.
For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
in :samp:`G` or zero otherwise. If the capacity is infinite,
:samp:`R[u][v]['capacity']` will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in
:samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
:samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.
The flow value, defined as the total flow into :samp:`t`, the sink, is
stored in :samp:`R.graph['flow_value']`. Reachability to :samp:`t` using
only edges :samp:`(u, v)` such that
:samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
:samp:`s`-:samp:`t` cut.
Specific algorithms may store extra data in :samp:`R`.
The function should supports an optional boolean parameter value_only. When
True, it can optionally terminate the algorithm as soon as the maximum flow
value and the minimum cut can be determined.
Examples
--------
>>> G = nx.DiGraph()
>>> G.add_edge("x", "a", capacity=3.0)
>>> G.add_edge("x", "b", capacity=1.0)
>>> G.add_edge("a", "c", capacity=3.0)
>>> G.add_edge("b", "c", capacity=5.0)
>>> G.add_edge("b", "d", capacity=4.0)
>>> G.add_edge("d", "e", capacity=2.0)
>>> G.add_edge("c", "y", capacity=2.0)
>>> G.add_edge("e", "y", capacity=3.0)
minimum_cut computes both the value of the
minimum cut and the node partition:
>>> cut_value, partition = nx.minimum_cut(G, "x", "y")
>>> reachable, non_reachable = partition
'partition' here is a tuple with the two sets of nodes that define
the minimum cut. You can compute the cut set of edges that induce
the minimum cut as follows:
>>> cutset = set()
>>> for u, nbrs in ((n, G[n]) for n in reachable):
... cutset.update((u, v) for v in nbrs if v in non_reachable)
>>> print(sorted(cutset))
[('c', 'y'), ('x', 'b')]
>>> cut_value == sum(G.edges[u, v]["capacity"] for (u, v) in cutset)
True
You can also use alternative algorithms for computing the
minimum cut by using the flow_func parameter.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> cut_value == nx.minimum_cut(G, "x", "y", flow_func=shortest_augmenting_path)[0]
True
"""
if flow_func is None:
if kwargs:
raise nx.NetworkXError(
"You have to explicitly set a flow_func if"
" you need to pass parameters via kwargs."
)
flow_func = default_flow_func
if not callable(flow_func):
raise nx.NetworkXError("flow_func has to be callable.")
if kwargs.get("cutoff") is not None and flow_func in flow_funcs:
raise nx.NetworkXError("cutoff should not be specified.")
R = flow_func(flowG, _s, _t, capacity=capacity, value_only=True, **kwargs)
# Remove saturated edges from the residual network
cutset = [(u, v, d) for u, v, d in R.edges(data=True) if d["flow"] == d["capacity"]]
R.remove_edges_from(cutset)
# Then, reachable and non reachable nodes from source in the
# residual network form the node partition that defines
# the minimum cut.
non_reachable = set(dict(nx.shortest_path_length(R, target=_t)))
partition = (set(flowG) - non_reachable, non_reachable)
# Finally add again cutset edges to the residual network to make
# sure that it is reusable.
if cutset is not None:
R.add_edges_from(cutset)
return (R.graph["flow_value"], partition)
[docs]def minimum_cut_value(flowG, _s, _t, capacity="capacity", flow_func=None, **kwargs):
"""Compute the value of a minimum (s, t)-cut.
Use the max-flow min-cut theorem, i.e., the capacity of a minimum
capacity cut is equal to the flow value of a maximum flow.
Parameters
----------
flowG : NetworkX graph
Edges of the graph are expected to have an attribute called
'capacity'. If this attribute is not present, the edge is
considered to have infinite capacity.
_s : node
Source node for the flow.
_t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: 'capacity'.
flow_func : function
A function for computing the maximum flow among a pair of nodes
in a capacitated graph. The function has to accept at least three
parameters: a Graph or Digraph, a source node, and a target node.
And return a residual network that follows NetworkX conventions
(see Notes). If flow_func is None, the default maximum
flow function (:meth:`preflow_push`) is used. See below for
alternative algorithms. The choice of the default function may change
from version to version and should not be relied on. Default value:
None.
kwargs : Any other keyword parameter is passed to the function that
computes the maximum flow.
Returns
-------
cut_value : integer, float
Value of the minimum cut.
Raises
------
NetworkXUnbounded
If the graph has a path of infinite capacity, all cuts have
infinite capacity and the function raises a NetworkXError.
See also
--------
:meth:`maximum_flow`
:meth:`maximum_flow_value`
:meth:`minimum_cut`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
Notes
-----
The function used in the flow_func parameter has to return a residual
network that follows NetworkX conventions:
The residual network :samp:`R` from an input graph :samp:`G` has the
same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
in :samp:`G`.
For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
in :samp:`G` or zero otherwise. If the capacity is infinite,
:samp:`R[u][v]['capacity']` will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in
:samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
:samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.
The flow value, defined as the total flow into :samp:`t`, the sink, is
stored in :samp:`R.graph['flow_value']`. Reachability to :samp:`t` using
only edges :samp:`(u, v)` such that
:samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
:samp:`s`-:samp:`t` cut.
Specific algorithms may store extra data in :samp:`R`.
The function should supports an optional boolean parameter value_only. When
True, it can optionally terminate the algorithm as soon as the maximum flow
value and the minimum cut can be determined.
Examples
--------
>>> G = nx.DiGraph()
>>> G.add_edge("x", "a", capacity=3.0)
>>> G.add_edge("x", "b", capacity=1.0)
>>> G.add_edge("a", "c", capacity=3.0)
>>> G.add_edge("b", "c", capacity=5.0)
>>> G.add_edge("b", "d", capacity=4.0)
>>> G.add_edge("d", "e", capacity=2.0)
>>> G.add_edge("c", "y", capacity=2.0)
>>> G.add_edge("e", "y", capacity=3.0)
minimum_cut_value computes only the value of the
minimum cut:
>>> cut_value = nx.minimum_cut_value(G, "x", "y")
>>> cut_value
3.0
You can also use alternative algorithms for computing the
minimum cut by using the flow_func parameter.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> cut_value == nx.minimum_cut_value(
... G, "x", "y", flow_func=shortest_augmenting_path
... )
True
"""
if flow_func is None:
if kwargs:
raise nx.NetworkXError(
"You have to explicitly set a flow_func if"
" you need to pass parameters via kwargs."
)
flow_func = default_flow_func
if not callable(flow_func):
raise nx.NetworkXError("flow_func has to be callable.")
if kwargs.get("cutoff") is not None and flow_func in flow_funcs:
raise nx.NetworkXError("cutoff should not be specified.")
R = flow_func(flowG, _s, _t, capacity=capacity, value_only=True, **kwargs)
return R.graph["flow_value"]