# This module uses material from the Wikipedia article Hopcroft--Karp algorithm
# <https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm>, accessed on
# January 3, 2015, which is released under the Creative Commons
# Attribution-Share-Alike License 3.0
# <http://creativecommons.org/licenses/by-sa/3.0/>. That article includes
# pseudocode, which has been translated into the corresponding Python code.
#
# Portions of this module use code from David Eppstein's Python Algorithms and
# Data Structures (PADS) library, which is dedicated to the public domain (for
# proof, see <http://www.ics.uci.edu/~eppstein/PADS/ABOUT-PADS.txt>).
"""Provides functions for computing maximum cardinality matchings and minimum
weight full matchings in a bipartite graph.
If you don't care about the particular implementation of the maximum matching
algorithm, simply use the :func:`maximum_matching`. If you do care, you can
import one of the named maximum matching algorithms directly.
For example, to find a maximum matching in the complete bipartite graph with
two vertices on the left and three vertices on the right:
>>> G = nx.complete_bipartite_graph(2, 3)
>>> left, right = nx.bipartite.sets(G)
>>> list(left)
[0, 1]
>>> list(right)
[2, 3, 4]
>>> nx.bipartite.maximum_matching(G)
{0: 2, 1: 3, 2: 0, 3: 1}
The dictionary returned by :func:`maximum_matching` includes a mapping for
vertices in both the left and right vertex sets.
Similarly, :func:`minimum_weight_full_matching` produces, for a complete
weighted bipartite graph, a matching whose cardinality is the cardinality of
the smaller of the two partitions, and for which the sum of the weights of the
edges included in the matching is minimal.
"""
import collections
import itertools
import networkx as nx
from networkx.algorithms.bipartite import sets as bipartite_sets
from networkx.algorithms.bipartite.matrix import biadjacency_matrix
__all__ = [
"maximum_matching",
"hopcroft_karp_matching",
"eppstein_matching",
"to_vertex_cover",
"minimum_weight_full_matching",
]
INFINITY = float("inf")
[docs]def hopcroft_karp_matching(G, top_nodes=None):
"""Returns the maximum cardinality matching of the bipartite graph `G`.
A matching is a set of edges that do not share any nodes. A maximum
cardinality matching is a matching with the most edges possible. It
is not always unique. Finding a matching in a bipartite graph can be
treated as a networkx flow problem.
The functions ``hopcroft_karp_matching`` and ``maximum_matching``
are aliases of the same function.
Parameters
----------
G : NetworkX graph
Undirected bipartite graph
top_nodes : container of nodes
Container with all nodes in one bipartite node set. If not supplied
it will be computed. But if more than one solution exists an exception
will be raised.
Returns
-------
matches : dictionary
The matching is returned as a dictionary, `matches`, such that
``matches[v] == w`` if node `v` is matched to node `w`. Unmatched
nodes do not occur as a key in `matches`.
Raises
------
AmbiguousSolution
Raised if the input bipartite graph is disconnected and no container
with all nodes in one bipartite set is provided. When determining
the nodes in each bipartite set more than one valid solution is
possible if the input graph is disconnected.
Notes
-----
This function is implemented with the `Hopcroft--Karp matching algorithm
<https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm>`_ for
bipartite graphs.
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
See Also
--------
maximum_matching
hopcroft_karp_matching
eppstein_matching
References
----------
.. [1] John E. Hopcroft and Richard M. Karp. "An n^{5 / 2} Algorithm for
Maximum Matchings in Bipartite Graphs" In: **SIAM Journal of Computing**
2.4 (1973), pp. 225--231. <https://doi.org/10.1137/0202019>.
"""
# First we define some auxiliary search functions.
#
# If you are a human reading these auxiliary search functions, the "global"
# variables `leftmatches`, `rightmatches`, `distances`, etc. are defined
# below the functions, so that they are initialized close to the initial
# invocation of the search functions.
def breadth_first_search():
for v in left:
if leftmatches[v] is None:
distances[v] = 0
queue.append(v)
else:
distances[v] = INFINITY
distances[None] = INFINITY
while queue:
v = queue.popleft()
if distances[v] < distances[None]:
for u in G[v]:
if distances[rightmatches[u]] is INFINITY:
distances[rightmatches[u]] = distances[v] + 1
queue.append(rightmatches[u])
return distances[None] is not INFINITY
def depth_first_search(v):
if v is not None:
for u in G[v]:
if distances[rightmatches[u]] == distances[v] + 1:
if depth_first_search(rightmatches[u]):
rightmatches[u] = v
leftmatches[v] = u
return True
distances[v] = INFINITY
return False
return True
# Initialize the "global" variables that maintain state during the search.
left, right = bipartite_sets(G, top_nodes)
leftmatches = {v: None for v in left}
rightmatches = {v: None for v in right}
distances = {}
queue = collections.deque()
# Implementation note: this counter is incremented as pairs are matched but
# it is currently not used elsewhere in the computation.
num_matched_pairs = 0
while breadth_first_search():
for v in left:
if leftmatches[v] is None:
if depth_first_search(v):
num_matched_pairs += 1
# Strip the entries matched to `None`.
leftmatches = {k: v for k, v in leftmatches.items() if v is not None}
rightmatches = {k: v for k, v in rightmatches.items() if v is not None}
# At this point, the left matches and the right matches are inverses of one
# another. In other words,
#
# leftmatches == {v, k for k, v in rightmatches.items()}
#
# Finally, we combine both the left matches and right matches.
return dict(itertools.chain(leftmatches.items(), rightmatches.items()))
[docs]def eppstein_matching(G, top_nodes=None):
"""Returns the maximum cardinality matching of the bipartite graph `G`.
Parameters
----------
G : NetworkX graph
Undirected bipartite graph
top_nodes : container
Container with all nodes in one bipartite node set. If not supplied
it will be computed. But if more than one solution exists an exception
will be raised.
Returns
-------
matches : dictionary
The matching is returned as a dictionary, `matching`, such that
``matching[v] == w`` if node `v` is matched to node `w`. Unmatched
nodes do not occur as a key in `matching`.
Raises
------
AmbiguousSolution
Raised if the input bipartite graph is disconnected and no container
with all nodes in one bipartite set is provided. When determining
the nodes in each bipartite set more than one valid solution is
possible if the input graph is disconnected.
Notes
-----
This function is implemented with David Eppstein's version of the algorithm
Hopcroft--Karp algorithm (see :func:`hopcroft_karp_matching`), which
originally appeared in the `Python Algorithms and Data Structures library
(PADS) <http://www.ics.uci.edu/~eppstein/PADS/ABOUT-PADS.txt>`_.
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
See Also
--------
hopcroft_karp_matching
"""
# Due to its original implementation, a directed graph is needed
# so that the two sets of bipartite nodes can be distinguished
left, right = bipartite_sets(G, top_nodes)
G = nx.DiGraph(G.edges(left))
# initialize greedy matching (redundant, but faster than full search)
matching = {}
for u in G:
for v in G[u]:
if v not in matching:
matching[v] = u
break
while True:
# structure residual graph into layers
# pred[u] gives the neighbor in the previous layer for u in U
# preds[v] gives a list of neighbors in the previous layer for v in V
# unmatched gives a list of unmatched vertices in final layer of V,
# and is also used as a flag value for pred[u] when u is in the first
# layer
preds = {}
unmatched = []
pred = {u: unmatched for u in G}
for v in matching:
del pred[matching[v]]
layer = list(pred)
# repeatedly extend layering structure by another pair of layers
while layer and not unmatched:
newLayer = {}
for u in layer:
for v in G[u]:
if v not in preds:
newLayer.setdefault(v, []).append(u)
layer = []
for v in newLayer:
preds[v] = newLayer[v]
if v in matching:
layer.append(matching[v])
pred[matching[v]] = v
else:
unmatched.append(v)
# did we finish layering without finding any alternating paths?
if not unmatched:
unlayered = {}
for u in G:
# TODO Why is extra inner loop necessary?
for v in G[u]:
if v not in preds:
unlayered[v] = None
# TODO Originally, this function returned a three-tuple:
#
# return (matching, list(pred), list(unlayered))
#
# For some reason, the documentation for this function
# indicated that the second and third elements of the returned
# three-tuple would be the vertices in the left and right vertex
# sets, respectively, that are also in the maximum independent set.
# However, what I think the author meant was that the second
# element is the list of vertices that were unmatched and the third
# element was the list of vertices that were matched. Since that
# seems to be the case, they don't really need to be returned,
# since that information can be inferred from the matching
# dictionary.
# All the matched nodes must be a key in the dictionary
for key in matching.copy():
matching[matching[key]] = key
return matching
# recursively search backward through layers to find alternating paths
# recursion returns true if found path, false otherwise
def recurse(v):
if v in preds:
L = preds.pop(v)
for u in L:
if u in pred:
pu = pred.pop(u)
if pu is unmatched or recurse(pu):
matching[v] = u
return True
return False
for v in unmatched:
recurse(v)
def _is_connected_by_alternating_path(G, v, matched_edges, unmatched_edges, targets):
"""Returns True if and only if the vertex `v` is connected to one of
the target vertices by an alternating path in `G`.
An *alternating path* is a path in which every other edge is in the
specified maximum matching (and the remaining edges in the path are not in
the matching). An alternating path may have matched edges in the even
positions or in the odd positions, as long as the edges alternate between
'matched' and 'unmatched'.
`G` is an undirected bipartite NetworkX graph.
`v` is a vertex in `G`.
`matched_edges` is a set of edges present in a maximum matching in `G`.
`unmatched_edges` is a set of edges not present in a maximum
matching in `G`.
`targets` is a set of vertices.
"""
def _alternating_dfs(u, along_matched=True):
"""Returns True if and only if `u` is connected to one of the
targets by an alternating path.
`u` is a vertex in the graph `G`.
If `along_matched` is True, this step of the depth-first search
will continue only through edges in the given matching. Otherwise, it
will continue only through edges *not* in the given matching.
"""
visited = set()
# Follow matched edges when depth is even,
# and follow unmatched edges when depth is odd.
initial_depth = 0 if along_matched else 1
stack = [(u, iter(G[u]), initial_depth)]
while stack:
parent, children, depth = stack[-1]
valid_edges = matched_edges if depth % 2 else unmatched_edges
try:
child = next(children)
if child not in visited:
if (parent, child) in valid_edges or (child, parent) in valid_edges:
if child in targets:
return True
visited.add(child)
stack.append((child, iter(G[child]), depth + 1))
except StopIteration:
stack.pop()
return False
# Check for alternating paths starting with edges in the matching, then
# check for alternating paths starting with edges not in the
# matching.
return _alternating_dfs(v, along_matched=True) or _alternating_dfs(
v, along_matched=False
)
def _connected_by_alternating_paths(G, matching, targets):
"""Returns the set of vertices that are connected to one of the target
vertices by an alternating path in `G` or are themselves a target.
An *alternating path* is a path in which every other edge is in the
specified maximum matching (and the remaining edges in the path are not in
the matching). An alternating path may have matched edges in the even
positions or in the odd positions, as long as the edges alternate between
'matched' and 'unmatched'.
`G` is an undirected bipartite NetworkX graph.
`matching` is a dictionary representing a maximum matching in `G`, as
returned by, for example, :func:`maximum_matching`.
`targets` is a set of vertices.
"""
# Get the set of matched edges and the set of unmatched edges. Only include
# one version of each undirected edge (for example, include edge (1, 2) but
# not edge (2, 1)). Using frozensets as an intermediary step we do not
# require nodes to be orderable.
edge_sets = {frozenset((u, v)) for u, v in matching.items()}
matched_edges = {tuple(edge) for edge in edge_sets}
unmatched_edges = {
(u, v) for (u, v) in G.edges() if frozenset((u, v)) not in edge_sets
}
return {
v
for v in G
if v in targets
or _is_connected_by_alternating_path(
G, v, matched_edges, unmatched_edges, targets
)
}
[docs]def to_vertex_cover(G, matching, top_nodes=None):
"""Returns the minimum vertex cover corresponding to the given maximum
matching of the bipartite graph `G`.
Parameters
----------
G : NetworkX graph
Undirected bipartite graph
matching : dictionary
A dictionary whose keys are vertices in `G` and whose values are the
distinct neighbors comprising the maximum matching for `G`, as returned
by, for example, :func:`maximum_matching`. The dictionary *must*
represent the maximum matching.
top_nodes : container
Container with all nodes in one bipartite node set. If not supplied
it will be computed. But if more than one solution exists an exception
will be raised.
Returns
-------
vertex_cover : :class:`set`
The minimum vertex cover in `G`.
Raises
------
AmbiguousSolution
Raised if the input bipartite graph is disconnected and no container
with all nodes in one bipartite set is provided. When determining
the nodes in each bipartite set more than one valid solution is
possible if the input graph is disconnected.
Notes
-----
This function is implemented using the procedure guaranteed by `Konig's
theorem
<https://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_%28graph_theory%29>`_,
which proves an equivalence between a maximum matching and a minimum vertex
cover in bipartite graphs.
Since a minimum vertex cover is the complement of a maximum independent set
for any graph, one can compute the maximum independent set of a bipartite
graph this way:
>>> G = nx.complete_bipartite_graph(2, 3)
>>> matching = nx.bipartite.maximum_matching(G)
>>> vertex_cover = nx.bipartite.to_vertex_cover(G, matching)
>>> independent_set = set(G) - vertex_cover
>>> print(list(independent_set))
[2, 3, 4]
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
"""
# This is a Python implementation of the algorithm described at
# <https://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_%28graph_theory%29#Proof>.
L, R = bipartite_sets(G, top_nodes)
# Let U be the set of unmatched vertices in the left vertex set.
unmatched_vertices = set(G) - set(matching)
U = unmatched_vertices & L
# Let Z be the set of vertices that are either in U or are connected to U
# by alternating paths.
Z = _connected_by_alternating_paths(G, matching, U)
# At this point, every edge either has a right endpoint in Z or a left
# endpoint not in Z. This gives us the vertex cover.
return (L - Z) | (R & Z)
#: Returns the maximum cardinality matching in the given bipartite graph.
#:
#: This function is simply an alias for :func:`hopcroft_karp_matching`.
maximum_matching = hopcroft_karp_matching
[docs]def minimum_weight_full_matching(G, top_nodes=None, weight="weight"):
r"""Returns a minimum weight full matching of the bipartite graph `G`.
Let :math:`G = ((U, V), E)` be a weighted bipartite graph with real weights
:math:`w : E \to \mathbb{R}`. This function then produces a matching
:math:`M \subseteq E` with cardinality
.. math::
\lvert M \rvert = \min(\lvert U \rvert, \lvert V \rvert),
which minimizes the sum of the weights of the edges included in the
matching, :math:`\sum_{e \in M} w(e)`, or raises an error if no such
matching exists.
When :math:`\lvert U \rvert = \lvert V \rvert`, this is commonly
referred to as a perfect matching; here, since we allow
:math:`\lvert U \rvert` and :math:`\lvert V \rvert` to differ, we
follow Karp [1]_ and refer to the matching as *full*.
Parameters
----------
G : NetworkX graph
Undirected bipartite graph
top_nodes : container
Container with all nodes in one bipartite node set. If not supplied
it will be computed.
weight : string, optional (default='weight')
The edge data key used to provide each value in the matrix.
Returns
-------
matches : dictionary
The matching is returned as a dictionary, `matches`, such that
``matches[v] == w`` if node `v` is matched to node `w`. Unmatched
nodes do not occur as a key in `matches`.
Raises
------
ValueError
Raised if no full matching exists.
ImportError
Raised if SciPy is not available.
Notes
-----
The problem of determining a minimum weight full matching is also known as
the rectangular linear assignment problem. This implementation defers the
calculation of the assignment to SciPy.
References
----------
.. [1] Richard Manning Karp:
An algorithm to Solve the m x n Assignment Problem in Expected Time
O(mn log n).
Networks, 10(2):143–152, 1980.
"""
import numpy as np
import scipy as sp
import scipy.optimize # call as sp.optimize
left, right = nx.bipartite.sets(G, top_nodes)
U = list(left)
V = list(right)
# We explicitly create the biadjancency matrix having infinities
# where edges are missing (as opposed to zeros, which is what one would
# get by using toarray on the sparse matrix).
weights_sparse = biadjacency_matrix(
G, row_order=U, column_order=V, weight=weight, format="coo"
)
weights = np.full(weights_sparse.shape, np.inf)
weights[weights_sparse.row, weights_sparse.col] = weights_sparse.data
left_matches = sp.optimize.linear_sum_assignment(weights)
d = {U[u]: V[v] for u, v in zip(*left_matches)}
# d will contain the matching from edges in left to right; we need to
# add the ones from right to left as well.
d.update({v: u for u, v in d.items()})
return d