#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
# Copyright (C) 2006-2025 Tiago de Paula Peixoto <tiago@skewed.de>
#
# This program is free software; you can redistribute it and/or modify it under
# the terms of the GNU Lesser General Public License as published by the Free
# Software Foundation; either version 3 of the License, or (at your option) any
# later version.
#
# This program is distributed in the hope that it will be useful, but WITHOUT
# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
# FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
# details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
"""
``graph_tool.flow``
-------------------
This module includes maximum flow / minimum cut algorithms.
Summary
+++++++
.. autosummary::
:nosignatures:
:toctree: autosummary
edmonds_karp_max_flow
push_relabel_max_flow
boykov_kolmogorov_max_flow
min_st_cut
min_cut
Example network
+++++++++++++++
The following network will be used as an example throughout the documentation.
.. testcode::
from numpy.random import seed, random
from scipy.linalg import norm
gt.seed_rng(42)
seed(42)
points = random((400, 2))
points[0] = [0, 0]
points[1] = [1, 1]
g, pos = gt.triangulation(points, type="delaunay")
g.set_directed(True)
edges = list(g.edges())
# reciprocate edges
for e in edges:
g.add_edge(e.target(), e.source())
# The capacity will be defined as the inverse euclidean distance
cap = g.new_edge_property("double")
for e in g.edges():
cap[e] = min((1.0 / norm(pos[e.target()].a - pos[e.source()].a), 10))
g.edge_properties["cap"] = cap
g.vertex_properties["pos"] = pos
g.save("flow-example.xml.gz")
gt.graph_draw(g, pos=pos, edge_pen_width=gt.prop_to_size(cap, mi=0, ma=3, power=1),
output="flow-example.pdf")
.. testcleanup::
conv_png("flow-example.pdf")
.. figure:: autosummary/flow-example.png
:align: center
:width: 80%
Example network used in the documentation below. The edge capacities are
represented by the edge width.
"""
from .. dl_import import dl_import
dl_import("from . import libgraph_tool_flow")
from .. import _prop, _check_prop_scalar, _check_prop_writable, GraphView
__all__ = ["edmonds_karp_max_flow", "push_relabel_max_flow",
"boykov_kolmogorov_max_flow", "min_st_cut", "min_cut"]
[docs]
def edmonds_karp_max_flow(g, source, target, capacity, residual=None):
r"""
Calculate maximum flow on the graph with the Edmonds-Karp algorithm.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
source : Vertex
The source vertex.
target : Vertex
The target (or "sink") vertex.
capacity : :class:`~graph_tool.EdgePropertyMap`
Edge property map with the edge capacities.
residual : :class:`~graph_tool.EdgePropertyMap` (optional, default: none)
Edge property map where the residuals should be stored.
Returns
-------
residual : :class:`~graph_tool.EdgePropertyMap`
Edge property map with the residual capacities (capacity - flow).
Notes
-----
The algorithm is due to [edmonds-theoretical-1972]_, though we are using the
variation called the "labeling algorithm" described in
[ravindra-network-1993]_.
This algorithm provides a very simple and easy to implement solution to the
maximum flow problem. However, there are several reasons why this algorithm
is not as good as the push_relabel_max_flow() or the
boykov_kolmogorov_max_flow() algorithm.
- In the non-integer capacity case, the time complexity is :math:`O(VE^2)`
which is worse than the time complexity of the push-relabel algorithm
:math:`O(V^2E^{1/2})` for all but the sparsest of graphs.
- In the integer capacity case, if the capacity bound U is very large then
the algorithm will take a long time.
The time complexity is :math:`O(VE^2)` in the general case or :math:`O(VEU)`
if capacity values are integers bounded by some constant :math:`U`.
Examples
--------
>>> g = gt.load_graph("flow-example.xml.gz")
>>> cap = g.edge_properties["cap"]
>>> src, tgt = g.vertex(0), g.vertex(1)
>>> res = gt.edmonds_karp_max_flow(g, src, tgt, cap)
>>> res.a = cap.a - res.a # the actual flow
>>> max_flow = sum(res[e] for e in tgt.in_edges())
>>> print(max_flow)
44.8905957841...
>>> pos = g.vertex_properties["pos"]
>>> gt.graph_draw(g, pos=pos, edge_pen_width=gt.prop_to_size(res, mi=0, ma=5, power=1), output="example-edmonds-karp.pdf")
<...>
.. testcleanup::
conv_png("example-edmonds-karp.pdf")
.. figure:: example-edmonds-karp.png
:align: center
:width: 80%
Edge flows obtained with the Edmonds-Karp algorithm. The source and
target are on the upper left and lower right corners, respectively. The
edge flows are represented by the edge width.
References
----------
.. [boost-edmonds-karp] http://www.boost.org/libs/graph/doc/edmonds_karp_max_flow.html
.. [edmonds-theoretical-1972] Jack Edmonds and Richard M. Karp, "Theoretical
improvements in the algorithmic efficiency for network flow problems.
Journal of the ACM", 19:248-264, 1972 :doi:`10.1145/321694.321699`
.. [ravindra-network-1993] Ravindra K. Ahuja and Thomas L. Magnanti and
James B. Orlin,"Network Flows: Theory, Algorithms, and Applications".
Prentice Hall, 1993.
"""
_check_prop_scalar(capacity, "capacity")
if residual is None:
residual = g.new_edge_property(capacity.value_type())
_check_prop_scalar(residual, "residual")
_check_prop_writable(residual, "residual")
if not g.is_directed():
raise ValueError("The graph provided must be directed!")
libgraph_tool_flow.\
edmonds_karp_max_flow(g._Graph__graph, int(source), int(target),
_prop("e", g, capacity),
_prop("e", g, residual))
return residual
[docs]
def push_relabel_max_flow(g, source, target, capacity, residual=None):
r"""
Calculate maximum flow on the graph with the push-relabel algorithm.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
source : Vertex
The source vertex.
target : Vertex
The target (or "sink") vertex.
capacity : :class:`~graph_tool.EdgePropertyMap`
Edge property map with the edge capacities.
residual : :class:`~graph_tool.EdgePropertyMap` (optional, default: none)
Edge property map where the residuals should be stored.
Returns
-------
residual : :class:`~graph_tool.EdgePropertyMap`
Edge property map with the residual capacities (capacity - flow).
Notes
-----
The algorithm is defined in [goldberg-new-1985]_. The complexity is
:math:`O(V^3)`.
Examples
--------
>>> g = gt.load_graph("flow-example.xml.gz")
>>> cap = g.edge_properties["cap"]
>>> src, tgt = g.vertex(0), g.vertex(1)
>>> res = gt.push_relabel_max_flow(g, src, tgt, cap)
>>> res.a = cap.a - res.a # the actual flow
>>> max_flow = sum(res[e] for e in tgt.in_edges())
>>> print(max_flow)
44.8905957841...
>>> pos = g.vertex_properties["pos"]
>>> gt.graph_draw(g, pos=pos, edge_pen_width=gt.prop_to_size(res, mi=0, ma=5, power=1), output="example-push-relabel.pdf")
<...>
.. testcleanup::
conv_png("example-push-relabel.pdf")
.. figure:: example-push-relabel.png
:align: center
:width: 80%
Edge flows obtained with the push-relabel algorithm. The source and
target are on the upper left and lower right corners, respectively. The
edge flows are represented by the edge width.
References
----------
.. [boost-push-relabel] http://www.boost.org/libs/graph/doc/push_relabel_max_flow.html
.. [goldberg-new-1985] A. V. Goldberg, "A New Max-Flow Algorithm", MIT
Tehnical report MIT/LCS/TM-291, 1985.
"""
_check_prop_scalar(capacity, "capacity")
if residual is None:
residual = g.new_edge_property(capacity.value_type())
_check_prop_scalar(residual, "residual")
_check_prop_writable(residual, "residual")
if not g.is_directed():
raise ValueError("The graph provided must be directed!")
libgraph_tool_flow.\
push_relabel_max_flow(g._Graph__graph, int(source), int(target),
_prop("e", g, capacity),
_prop("e", g, residual))
return residual
[docs]
def boykov_kolmogorov_max_flow(g, source, target, capacity, residual=None):
r"""Calculate maximum flow on the graph with the Boykov-Kolmogorov algorithm.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
source : Vertex
The source vertex.
target : Vertex
The target (or "sink") vertex.
capacity : :class:`~graph_tool.EdgePropertyMap`
Edge property map with the edge capacities.
residual : :class:`~graph_tool.EdgePropertyMap` (optional, default: none)
Edge property map where the residuals should be stored.
Returns
-------
residual : :class:`~graph_tool.EdgePropertyMap`
Edge property map with the residual capacities (capacity - flow).
Notes
-----
The algorithm is defined in [kolmogorov-graph-2003]_ and
[boykov-experimental-2004]_. The worst case complexity is
:math:`O(EV^2|C|)`, where :math:`|C|` is the minimum cut (but typically
performs much better).
For a more detailed description, see [boost-kolmogorov]_.
Examples
--------
>>> g = gt.load_graph("flow-example.xml.gz")
>>> cap = g.edge_properties["cap"]
>>> src, tgt = g.vertex(0), g.vertex(1)
>>> res = gt.boykov_kolmogorov_max_flow(g, src, tgt, cap)
>>> res.a = cap.a - res.a # the actual flow
>>> max_flow = sum(res[e] for e in tgt.in_edges())
>>> print(max_flow)
44.8905957841...
>>> pos = g.vertex_properties["pos"]
>>> gt.graph_draw(g, pos=pos, edge_pen_width=gt.prop_to_size(res, mi=0, ma=3, power=1), output="example-kolmogorov.pdf")
<...>
.. testcleanup::
conv_png("example-kolmogorov.pdf")
.. figure:: example-kolmogorov.png
:align: center
:width: 80%
Edge flows obtained with the Boykov-Kolmogorov algorithm. The source and
target are on the upper left and lower right corners, respectively. The
edge flows are represented by the edge width.
References
----------
.. [boost-kolmogorov] http://www.boost.org/libs/graph/doc/boykov_kolmogorov_max_flow.html
.. [kolmogorov-graph-2003] Vladimir Kolmogorov, "Graph Based Algorithms for
Scene Reconstruction from Two or More Views", PhD thesis, Cornell
University, September 2003.
.. [boykov-experimental-2004] Yuri Boykov and Vladimir Kolmogorov, "An
Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy
Minimization in Vision", IEEE Transactions on Pattern Analysis and
Machine Intelligence, vol. 26, no. 9, pp. 1124-1137, Sept. 2004.
:doi:`10.1109/TPAMI.2004.60`
"""
_check_prop_scalar(capacity, "capacity")
if residual is None:
residual = g.new_edge_property(capacity.value_type())
_check_prop_scalar(residual, "residual")
_check_prop_writable(residual, "residual")
if not g.is_directed():
raise ValueError("The graph provided must be directed!")
libgraph_tool_flow.\
kolmogorov_max_flow(g._Graph__graph, int(source), int(target),
_prop("e", g, capacity),
_prop("e", g, residual))
return residual
[docs]
def min_st_cut(g, source, capacity, residual):
r"""
Get the minimum source-target cut, given the residual capacity of the edges.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
source : Vertex
The source vertex.
capacity : :class:`~graph_tool.EdgePropertyMap`
Edge property map with the edge capacities.
residual : :class:`~graph_tool.EdgePropertyMap`
Edge property map with the residual capacities (capacity - flow).
Returns
-------
partition : :class:`~graph_tool.EdgePropertyMap`
Boolean-valued vertex property map with the cut partition. Vertices with
value `True` belong to the source side of the cut.
Notes
-----
The source-side of the cut set is obtained by following all vertices which
are reachable from the source in the residual graph (i.e. via edges
with nonzero residual capacity, and reversed edges with nonzero flow).
This algorithm runs in :math:`O(V+E)` time.
Examples
--------
>>> g = gt.load_graph("../mincut-st-example.xml.gz")
>>> cap = g.edge_properties["weight"]
>>> src, tgt = g.vertex(0), g.vertex(7)
>>> res = gt.boykov_kolmogorov_max_flow(g, src, tgt, cap)
>>> part = gt.min_st_cut(g, src, cap, res)
>>> mc = sum([cap[e] - res[e] for e in g.edges() if part[e.source()] != part[e.target()]])
>>> print(mc)
3
>>> pos = g.vertex_properties["pos"]
>>> res.a = cap.a - res.a # the actual flow
>>> gt.graph_draw(g, pos=pos, edge_pen_width=gt.prop_to_size(cap, mi=3, ma=10, power=1),
... edge_text=res, vertex_fill_color=part, vertex_text=g.vertex_index,
... vertex_font_size=18, edge_font_size=18, output="example-min-st-cut.pdf")
<...>
.. testcleanup::
conv_png("example-min-st-cut.pdf")
.. figure:: example-min-st-cut.png
:align: center
:width: 30%
Edge flows obtained with the Boykov-Kolmogorov algorithm. The source and
target are labeled ``0`` and ``7``, respectively. The edge capacities are
represented by the edge width, and the maximum flow by the edge labels.
Vertices of the same color are on the same side the minimum cut.
References
----------
.. [max-flow-min-cut] http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem
"""
if not g.is_directed():
raise ValueError("The graph provided must be directed!")
augment = g.new_edge_property("bool")
libgraph_tool_flow.residual_graph(g._Graph__graph,
_prop("e", g, capacity),
_prop("e", g, residual),
_prop("e", g, augment))
em = g.new_edge_property("bool")
em.a = (residual.a[:len(em.a)] > 0) + augment.a
u = GraphView(g, efilt=em)
part = label_out_component(u, source)
part = g.own_property(part)
# cleanup augmented edges
u = GraphView(g, efilt=augment)
u.clear_edges()
return part
[docs]
def min_cut(g, weight):
r"""
Get the minimum cut of an undirected graph, given the weight of the edges.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
weight : :class:`~graph_tool.EdgePropertyMap`
Edge property map with the edge weights.
Returns
-------
min_cut : float
The value of the minimum cut.
partition : :class:`~graph_tool.VertexPropertyMap`
Boolean-valued vertex property map with the cut partition.
Notes
-----
The algorithm is defined in [stoer_simple_1997]_.
The time complexity is :math:`O(VE + V^2 \log V)`.
Examples
--------
>>> g = gt.load_graph("../mincut-example.xml.gz")
>>> weight = g.edge_properties["weight"]
>>> mc, part = gt.min_cut(g, weight)
>>> print(mc)
4.0
>>> pos = g.vertex_properties["pos"]
>>> gt.graph_draw(g, pos=pos, edge_pen_width=weight, vertex_fill_color=part,
... output="example-min-cut.pdf")
<...>
.. testcleanup::
conv_png("example-min-cut.pdf")
.. figure:: example-min-cut.png
:align: center
:width: 30%
Vertices of the same color are on the same side of a minimum cut. The
edge weights are represented by the edge width.
References
----------
.. [stoer_simple_1997] Stoer, Mechthild and Frank Wagner, "A simple min-cut
algorithm". Journal of the ACM 44 (4), 585-591, 1997. :doi:`10.1145/263867.263872`
"""
_check_prop_scalar(weight, "weight")
if g.is_directed():
raise ValueError("The graph provided must be undirected!")
part = g.new_vertex_property("bool")
mc = libgraph_tool_flow.min_cut(g._Graph__graph,
_prop("e", g, weight),
_prop("v", g, part))
return mc, part
from .. topology import label_out_component
