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