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"""
********
Matching
********
"""
# Copyright (C) 20042008 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
# Copyright (C) 2011 by
# Nicholas Mancuso <nick.mancuso@gmail.com>
# All rights reserved.
# BSD license.
from itertools import repeat
__author__ = """\n""".join(['Joris van Rantwijk',
'Nicholas Mancuso (nick.mancuso@gmail.com)'])
_all__ = ['max_weight_matching', 'maximal_matching']
def maximal_matching(graph):
""" Find a maximal matching in the graph.
Parameters

graph : NetworkX graph
Undirected graph
Returns

matching : set of edges.
A maximal mathing of the graph.
"""
matching = set([])
edges = set([])
for edge in graph.edges_iter():
if edge not in edges:
u, v = edge
matching.add(edge)
edges = edges  set(graph.edges(u))  set(graph.edges(v))
return matching
def max_weight_matching(G, maxcardinality=False):
"""Compute a maximumweighted matching of G.
A matching is a subset of edges in which no node occurs more than once.
The cardinality of a matching is the number of matched edges.
The weight of a matching is the sum of the weights of its edges.
Parameters

G : NetworkX graph
Undirected graph
maxcardinality: bool, optional
If maxcardinality is True, compute the maximumcardinality matching
with maximum weight among all maximumcardinality matchings.
Returns

mate : dictionary
The matching is returned as a dictionary, mate, such that
mate[v] == w if node v is matched to node w. Unmatched nodes do not
occur as a key in mate.
Notes

If G has edges with 'weight' attribute the edge data are used as
weight values else the weights are assumed to be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer
computations. If floating point weights are used, the algorithm
could return a slightly suboptimal matching due to numeric
precision errors.
This method is based on the "blossom" method for finding augmenting
paths and the "primaldual" method for finding a matching of maximum
weight, both methods invented by Jack Edmonds [1]_.
References

.. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs",
Zvi Galil, ACM Computing Surveys, 1986.
"""
#
# The algorithm is taken from "Efficient Algorithms for Finding Maximum
# Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
# It is based on the "blossom" method for finding augmenting paths and
# the "primaldual" method for finding a matching of maximum weight, both
# methods invented by Jack Edmonds.
#
# A C program for maximum weight matching by Ed Rothberg was used
# extensively to validate this new code.
#
# Many terms used in the code comments are explained in the paper
# by Galil. You will probably need the paper to make sense of this code.
#
class NoNode:
"""Dummy value which is different from any node."""
pass
class Blossom:
"""Representation of a nontrivial blossom or subblossom."""
__slots__ = [ 'childs', 'edges', 'mybestedges' ]
# b.childs is an ordered list of b's subblossoms, starting with
# the base and going round the blossom.
# b.edges is the list of b's connecting edges, such that
# b.edges[i] = (v, w) where v is a vertex in b.childs[i]
# and w is a vertex in b.childs[wrap(i+1)].
# If b is a toplevel Sblossom,
# b.mybestedges is a list of leastslack edges to neighbouring
# Sblossoms, or None if no such list has been computed yet.
# This is used for efficient computation of delta3.
# Generate the blossom's leaf vertices.
def leaves(self):
for t in self.childs:
if isinstance(t, Blossom):
for v in t.leaves():
yield v
else:
yield t
# Get a list of vertices.
gnodes = G.nodes()
if not gnodes:
return { } # don't bother with empty graphs
# Find the maximum edge weight.
maxweight = 0
allinteger = True
for i,j,d in G.edges_iter(data=True):
wt=d.get('weight',1)
if i != j and wt > maxweight:
maxweight = wt
allinteger = allinteger and (str(type(wt)).split("'")[1]
in ('int', 'long'))
# If v is a matched vertex, mate[v] is its partner vertex.
# If v is a single vertex, v does not occur as a key in mate.
# Initially all vertices are single; updated during augmentation.
mate = { }
# If b is a toplevel blossom,
# label.get(b) is None if b is unlabeled (free),
# 1 if b is an Sblossom,
# 2 if b is a Tblossom.
# The label of a vertex is found by looking at the label of its toplevel
# containing blossom.
# If v is a vertex inside a Tblossom, label[v] is 2 iff v is reachable
# from an Svertex outside the blossom.
# Labels are assigned during a stage and reset after each augmentation.
label = { }
# If b is a labeled toplevel blossom,
# labeledge[b] = (v, w) is the edge through which b obtained its label
# such that w is a vertex in b, or None if b's base vertex is single.
# If w is a vertex inside a Tblossom and label[w] == 2,
# labeledge[w] = (v, w) is an edge through which w is reachable from
# outside the blossom.
labeledge = { }
# If v is a vertex, inblossom[v] is the toplevel blossom to which v
# belongs.
# If v is a toplevel vertex, inblossom[v] == v since v is itself
# a (trivial) toplevel blossom.
# Initially all vertices are toplevel trivial blossoms.
inblossom = dict(zip(gnodes, gnodes))
# If b is a subblossom,
# blossomparent[b] is its immediate parent (sub)blossom.
# If b is a toplevel blossom, blossomparent[b] is None.
blossomparent = dict(zip(gnodes, repeat(None)))
# If b is a (sub)blossom,
# blossombase[b] is its base VERTEX (i.e. recursive subblossom).
blossombase = dict(zip(gnodes, gnodes))
# If w is a free vertex (or an unreached vertex inside a Tblossom),
# bestedge[w] = (v, w) is the leastslack edge from an Svertex,
# or None if there is no such edge.
# If b is a (possibly trivial) toplevel Sblossom,
# bestedge[b] = (v, w) is the leastslack edge to a different Sblossom
# (v inside b), or None if there is no such edge.
# This is used for efficient computation of delta2 and delta3.
bestedge = { }
# If v is a vertex,
# dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual
# optimization problem (if all edge weights are integers, multiplication
# by two ensures that all values remain integers throughout the algorithm).
# Initially, u(v) = maxweight / 2.
dualvar = dict(zip(gnodes, repeat(maxweight)))
# If b is a nontrivial blossom,
# blossomdual[b] = z(b) where z(b) is b's variable in the dual
# optimization problem.
blossomdual = { }
# If (v, w) in allowedge or (w, v) in allowedg, then the edge
# (v, w) is known to have zero slack in the optimization problem;
# otherwise the edge may or may not have zero slack.
allowedge = { }
# Queue of newly discovered Svertices.
queue = [ ]
# Return 2 * slack of edge (v, w) (does not work inside blossoms).
def slack(v, w):
return dualvar[v] + dualvar[w]  2 * G[v][w].get('weight',1)
# Assign label t to the toplevel blossom containing vertex w,
# coming through an edge from vertex v.
def assignLabel(w, t, v):
b = inblossom[w]
assert label.get(w) is None and label.get(b) is None
label[w] = label[b] = t
if v is not None:
labeledge[w] = labeledge[b] = (v, w)
else:
labeledge[w] = labeledge[b] = None
bestedge[w] = bestedge[b] = None
if t == 1:
# b became an Svertex/blossom; add it(s vertices) to the queue.
if isinstance(b, Blossom):
queue.extend(b.leaves())
else:
queue.append(b)
elif t == 2:
# b became a Tvertex/blossom; assign label S to its mate.
# (If b is a nontrivial blossom, its base is the only vertex
# with an external mate.)
base = blossombase[b]
assignLabel(mate[base], 1, base)
# Trace back from vertices v and w to discover either a new blossom
# or an augmenting path. Return the base vertex of the new blossom,
# or NoNode if an augmenting path was found.
def scanBlossom(v, w):
# Trace back from v and w, placing breadcrumbs as we go.
path = [ ]
base = NoNode
while v is not NoNode:
# Look for a breadcrumb in v's blossom or put a new breadcrumb.
b = inblossom[v]
if label[b] & 4:
base = blossombase[b]
break
assert label[b] == 1
path.append(b)
label[b] = 5
# Trace one step back.
if labeledge[b] is None:
# The base of blossom b is single; stop tracing this path.
assert blossombase[b] not in mate
v = NoNode
else:
assert labeledge[b][0] == mate[blossombase[b]]
v = labeledge[b][0]
b = inblossom[v]
assert label[b] == 2
# b is a Tblossom; trace one more step back.
v = labeledge[b][0]
# Swap v and w so that we alternate between both paths.
if w is not NoNode:
v, w = w, v
# Remove breadcrumbs.
for b in path:
label[b] = 1
# Return base vertex, if we found one.
return base
# Construct a new blossom with given base, through Svertices v and w.
# Label the new blossom as S; set its dual variable to zero;
# relabel its Tvertices to S and add them to the queue.
def addBlossom(base, v, w):
bb = inblossom[base]
bv = inblossom[v]
bw = inblossom[w]
# Create blossom.
b = Blossom()
blossombase[b] = base
blossomparent[b] = None
blossomparent[bb] = b
# Make list of subblossoms and their interconnecting edge endpoints.
b.childs = path = [ ]
b.edges = edgs = [ (v, w) ]
# Trace back from v to base.
while bv != bb:
# Add bv to the new blossom.
blossomparent[bv] = b
path.append(bv)
edgs.append(labeledge[bv])
assert label[bv] == 2 or (label[bv] == 1 and labeledge[bv][0] == mate[blossombase[bv]])
# Trace one step back.
v = labeledge[bv][0]
bv = inblossom[v]
# Add base subblossom; reverse lists.
path.append(bb)
path.reverse()
edgs.reverse()
# Trace back from w to base.
while bw != bb:
# Add bw to the new blossom.
blossomparent[bw] = b
path.append(bw)
edgs.append((labeledge[bw][1], labeledge[bw][0]))
assert label[bw] == 2 or (label[bw] == 1 and labeledge[bw][0] == mate[blossombase[bw]])
# Trace one step back.
w = labeledge[bw][0]
bw = inblossom[w]
# Set label to S.
assert label[bb] == 1
label[b] = 1
labeledge[b] = labeledge[bb]
# Set dual variable to zero.
blossomdual[b] = 0
# Relabel vertices.
for v in b.leaves():
if label[inblossom[v]] == 2:
# This Tvertex now turns into an Svertex because it becomes
# part of an Sblossom; add it to the queue.
queue.append(v)
inblossom[v] = b
# Compute b.mybestedges.
bestedgeto = { }
for bv in path:
if isinstance(bv, Blossom):
if bv.mybestedges is not None:
# Walk this subblossom's leastslack edges.
nblist = bv.mybestedges
# The subblossom won't need this data again.
bv.mybestedges = None
else:
# This subblossom does not have a list of leastslack
# edges; get the information from the vertices.
nblist = [ (v, w)
for v in bv.leaves()
for w in G.neighbors_iter(v)
if v != w ]
else:
nblist = [ (bv, w)
for w in G.neighbors_iter(bv)
if bv != w ]
for k in nblist:
(i, j) = k
if inblossom[j] == b:
i, j = j, i
bj = inblossom[j]
if (bj != b and label.get(bj) == 1 and
((bj not in bestedgeto) or
slack(i, j) < slack(*bestedgeto[bj]))):
bestedgeto[bj] = k
# Forget about leastslack edge of the subblossom.
bestedge[bv] = None
b.mybestedges = list(bestedgeto.values())
# Select bestedge[b].
mybestedge = None
bestedge[b] = None
for k in b.mybestedges:
kslack = slack(*k)
if mybestedge is None or kslack < mybestslack:
mybestedge = k
mybestslack = kslack
bestedge[b] = mybestedge
# Expand the given toplevel blossom.
def expandBlossom(b, endstage):
# Convert subblossoms into toplevel blossoms.
for s in b.childs:
blossomparent[s] = None
if isinstance(s, Blossom):
if endstage and blossomdual[s] == 0:
# Recursively expand this subblossom.
expandBlossom(s, endstage)
else:
for v in s.leaves():
inblossom[v] = s
else:
inblossom[s] = s
# If we expand a Tblossom during a stage, its subblossoms must be
# relabeled.
if (not endstage) and label.get(b) == 2:
# Start at the subblossom through which the expanding
# blossom obtained its label, and relabel subblossoms untili
# we reach the base.
# Figure out through which subblossom the expanding blossom
# obtained its label initially.
entrychild = inblossom[labeledge[b][1]]
# Decide in which direction we will go round the blossom.
j = b.childs.index(entrychild)
if j & 1:
# Start index is odd; go forward and wrap.
j = len(b.childs)
jstep = 1
else:
# Start index is even; go backward.
jstep = 1
# Move along the blossom until we get to the base.
v, w = labeledge[b]
while j != 0:
# Relabel the Tsubblossom.
if jstep == 1:
p, q = b.edges[j]
else:
q, p = b.edges[j1]
label[w] = None
label[q] = None
assignLabel(w, 2, v)
# Step to the next Ssubblossom and note its forward edge.
allowedge[(p, q)] = allowedge[(q, p)] = True
j += jstep
if jstep == 1:
v, w = b.edges[j]
else:
w, v = b.edges[j1]
# Step to the next Tsubblossom.
allowedge[(v, w)] = allowedge[(w, v)] = True
j += jstep
# Relabel the base Tsubblossom WITHOUT stepping through to
# its mate (so don't call assignLabel).
bw = b.childs[j]
label[w] = label[bw] = 2
labeledge[w] = labeledge[bw] = (v, w)
bestedge[bw] = None
# Continue along the blossom until we get back to entrychild.
j += jstep
while b.childs[j] != entrychild:
# Examine the vertices of the subblossom to see whether
# it is reachable from a neighbouring Svertex outside the
# expanding blossom.
bv = b.childs[j]
if label.get(bv) == 1:
# This subblossom just got label S through one of its
# neighbours; leave it be.
j += jstep
continue
if isinstance(bv, Blossom):
for v in bv.leaves():
if label.get(v):
break
else:
v = bv
# If the subblossom contains a reachable vertex, assign
# label T to the subblossom.
if label.get(v):
assert label[v] == 2
assert inblossom[v] == bv
label[v] = None
label[mate[blossombase[bv]]] = None
assignLabel(v, 2, labeledge[v][0])
j += jstep
# Remove the expanded blossom entirely.
label.pop(b, None)
labeledge.pop(b, None)
bestedge.pop(b, None)
del blossomparent[b]
del blossombase[b]
del blossomdual[b]
# Swap matched/unmatched edges over an alternating path through blossom b
# between vertex v and the base vertex. Keep blossom bookkeeping consistent.
def augmentBlossom(b, v):
# Bubble up through the blossom tree from vertex v to an immediate
# subblossom of b.
t = v
while blossomparent[t] != b:
t = blossomparent[t]
# Recursively deal with the first subblossom.
if isinstance(t, Blossom):
augmentBlossom(t, v)
# Decide in which direction we will go round the blossom.
i = j = b.childs.index(t)
if i & 1:
# Start index is odd; go forward and wrap.
j = len(b.childs)
jstep = 1
else:
# Start index is even; go backward.
jstep = 1
# Move along the blossom until we get to the base.
while j != 0:
# Step to the next subblossom and augment it recursively.
j += jstep
t = b.childs[j]
if jstep == 1:
w, x = b.edges[j]
else:
x, w = b.edges[j1]
if isinstance(t, Blossom):
augmentBlossom(t, w)
# Step to the next subblossom and augment it recursively.
j += jstep
t = b.childs[j]
if isinstance(t, Blossom):
augmentBlossom(t, x)
# Match the edge connecting those subblossoms.
mate[w] = x
mate[x] = w
# Rotate the list of subblossoms to put the new base at the front.
b.childs = b.childs[i:] + b.childs[:i]
b.edges = b.edges[i:] + b.edges[:i]
blossombase[b] = blossombase[b.childs[0]]
assert blossombase[b] == v
# Swap matched/unmatched edges over an alternating path between two
# single vertices. The augmenting path runs through Svertices v and w.
def augmentMatching(v, w):
for (s, j) in ((v, w), (w, v)):
# Match vertex s to vertex j. Then trace back from s
# until we find a single vertex, swapping matched and unmatched
# edges as we go.
while 1:
bs = inblossom[s]
assert label[bs] == 1
assert (labeledge[bs] is None and blossombase[bs] not in mate) or (labeledge[bs][0] == mate[blossombase[bs]])
# Augment through the Sblossom from s to base.
if isinstance(bs, Blossom):
augmentBlossom(bs, s)
# Update mate[s]
mate[s] = j
# Trace one step back.
if labeledge[bs] is None:
# Reached single vertex; stop.
break
t = labeledge[bs][0]
bt = inblossom[t]
assert label[bt] == 2
# Trace one more step back.
s, j = labeledge[bt]
# Augment through the Tblossom from j to base.
assert blossombase[bt] == t
if isinstance(bt, Blossom):
augmentBlossom(bt, j)
# Update mate[j]
mate[j] = s
# Verify that the optimum solution has been reached.
def verifyOptimum():
if maxcardinality:
# Vertices may have negative dual;
# find a constant nonnegative number to add to all vertex duals.
vdualoffset = max(0, min(dualvar.values()))
else:
vdualoffset = 0
# 0. all dual variables are nonnegative
assert min(dualvar.values()) + vdualoffset >= 0
assert len(blossomdual) == 0 or min(blossomdual.values()) >= 0
# 0. all edges have nonnegative slack and
# 1. all matched edges have zero slack;
for i,j,d in G.edges_iter(data=True):
wt=d.get('weight',1)
if i == j:
continue # ignore selfloops
s = dualvar[i] + dualvar[j]  2 * wt
iblossoms = [ i ]
jblossoms = [ j ]
while blossomparent[iblossoms[1]] is not None:
iblossoms.append(blossomparent[iblossoms[1]])
while blossomparent[jblossoms[1]] is not None:
jblossoms.append(blossomparent[jblossoms[1]])
iblossoms.reverse()
jblossoms.reverse()
for (bi, bj) in zip(iblossoms, jblossoms):
if bi != bj:
break
s += 2 * blossomdual[bi]
assert s >= 0
if mate.get(i) == j or mate.get(j) == i:
assert mate[i] == j and mate[j] == i
assert s == 0
# 2. all single vertices have zero dual value;
for v in gnodes:
assert (v in mate) or dualvar[v] + vdualoffset == 0
# 3. all blossoms with positive dual value are full.
for b in blossomdual:
if blossomdual[b] > 0:
assert len(b.edges) % 2 == 1
for (i, j) in b.edges[1::2]:
assert mate[i] == j and mate[j] == i
# Ok.
# Main loop: continue until no further improvement is possible.
while 1:
# Each iteration of this loop is a "stage".
# A stage finds an augmenting path and uses that to improve
# the matching.
# Remove labels from toplevel blossoms/vertices.
label.clear()
labeledge.clear()
# Forget all about leastslack edges.
bestedge.clear()
for b in blossomdual:
b.mybestedges = None
# Loss of labeling means that we can not be sure that currently
# allowable edges remain allowable througout this stage.
allowedge.clear()
# Make queue empty.
queue[:] = [ ]
# Label single blossoms/vertices with S and put them in the queue.
for v in gnodes:
if (v not in mate) and label.get(inblossom[v]) is None:
assignLabel(v, 1, None)
# Loop until we succeed in augmenting the matching.
augmented = 0
while 1:
# Each iteration of this loop is a "substage".
# A substage tries to find an augmenting path;
# if found, the path is used to improve the matching and
# the stage ends. If there is no augmenting path, the
# primaldual method is used to pump some slack out of
# the dual variables.
# Continue labeling until all vertices which are reachable
# through an alternating path have got a label.
while queue and not augmented:
# Take an S vertex from the queue.
v = queue.pop()
assert label[inblossom[v]] == 1
# Scan its neighbours:
for w in G.neighbors_iter(v):
if w == v:
continue # ignore selfloops
# w is a neighbour to v
bv = inblossom[v]
bw = inblossom[w]
if bv == bw:
# this edge is internal to a blossom; ignore it
continue
if (v, w) not in allowedge:
kslack = slack(v, w)
if kslack <= 0:
# edge k has zero slack => it is allowable
allowedge[(v, w)] = allowedge[(w, v)] = True
if (v, w) in allowedge:
if label.get(bw) is None:
# (C1) w is a free vertex;
# label w with T and label its mate with S (R12).
assignLabel(w, 2, v)
elif label.get(bw) == 1:
# (C2) w is an Svertex (not in the same blossom);
# follow backlinks to discover either an
# augmenting path or a new blossom.
base = scanBlossom(v, w)
if base is not NoNode:
# Found a new blossom; add it to the blossom
# bookkeeping and turn it into an Sblossom.
addBlossom(base, v, w)
else:
# Found an augmenting path; augment the
# matching and end this stage.
augmentMatching(v, w)
augmented = 1
break
elif label.get(w) is None:
# w is inside a Tblossom, but w itself has not
# yet been reached from outside the blossom;
# mark it as reached (we need this to relabel
# during Tblossom expansion).
assert label[bw] == 2
label[w] = 2
labeledge[w] = (v, w)
elif label.get(bw) == 1:
# keep track of the leastslack nonallowable edge to
# a different Sblossom.
if bestedge.get(bv) is None or kslack < slack(*bestedge[bv]):
bestedge[bv] = (v, w)
elif label.get(w) is None:
# w is a free vertex (or an unreached vertex inside
# a Tblossom) but we can not reach it yet;
# keep track of the leastslack edge that reaches w.
if bestedge.get(w) is None or kslack < slack(*bestedge[w]):
bestedge[w] = (v, w)
if augmented:
break
# There is no augmenting path under these constraints;
# compute delta and reduce slack in the optimization problem.
# (Note that our vertex dual variables, edge slacks and delta's
# are premultiplied by two.)
deltatype = 1
delta = deltaedge = deltablossom = None
# Compute delta1: the minumum value of any vertex dual.
if not maxcardinality:
deltatype = 1
delta = min(dualvar.values())
# Compute delta2: the minimum slack on any edge between
# an Svertex and a free vertex.
for v in G.nodes_iter():
if label.get(inblossom[v]) is None and bestedge.get(v) is not None:
d = slack(*bestedge[v])
if deltatype == 1 or d < delta:
delta = d
deltatype = 2
deltaedge = bestedge[v]
# Compute delta3: half the minimum slack on any edge between
# a pair of Sblossoms.
for b in blossomparent:
if ( blossomparent[b] is None and label.get(b) == 1 and
bestedge.get(b) is not None ):
kslack = slack(*bestedge[b])
if allinteger:
assert (kslack % 2) == 0
d = kslack // 2
else:
d = kslack / 2.0
if deltatype == 1 or d < delta:
delta = d
deltatype = 3
deltaedge = bestedge[b]
# Compute delta4: minimum z variable of any Tblossom.
for b in blossomdual:
if ( blossomparent[b] is None and label.get(b) == 2 and
(deltatype == 1 or blossomdual[b] < delta) ):
delta = blossomdual[b]
deltatype = 4
deltablossom = b
if deltatype == 1:
# No further improvement possible; maxcardinality optimum
# reached. Do a final delta update to make the optimum
# verifyable.
assert maxcardinality
deltatype = 1
delta = max(0, min(dualvar.values()))
# Update dual variables according to delta.
for v in gnodes:
if label.get(inblossom[v]) == 1:
# Svertex: 2*u = 2*u  2*delta
dualvar[v] = delta
elif label.get(inblossom[v]) == 2:
# Tvertex: 2*u = 2*u + 2*delta
dualvar[v] += delta
for b in blossomdual:
if blossomparent[b] is None:
if label.get(b) == 1:
# toplevel Sblossom: z = z + 2*delta
blossomdual[b] += delta
elif label.get(b) == 2:
# toplevel Tblossom: z = z  2*delta
blossomdual[b] = delta
# Take action at the point where minimum delta occurred.
if deltatype == 1:
# No further improvement possible; optimum reached.
break
elif deltatype == 2:
# Use the leastslack edge to continue the search.
(v, w) = deltaedge
assert label[inblossom[v]] == 1
allowedge[(v, w)] = allowedge[(w, v)] = True
queue.append(v)
elif deltatype == 3:
# Use the leastslack edge to continue the search.
(v, w) = deltaedge
allowedge[(v, w)] = allowedge[(w, v)] = True
assert label[inblossom[v]] == 1
queue.append(v)
elif deltatype == 4:
# Expand the leastz blossom.
expandBlossom(deltablossom, False)
# End of a this substage.
# Paranoia check that the matching is symmetric.
for v in mate:
assert mate[mate[v]] == v
# Stop when no more augmenting path can be found.
if not augmented:
break
# End of a stage; expand all Sblossoms which have zero dual.
for b in list(blossomdual.keys()):
if b not in blossomdual:
continue # already expanded
if ( blossomparent[b] is None and label.get(b) == 1 and
blossomdual[b] == 0 ):
expandBlossom(b, True)
# Verify that we reached the optimum solution (only for integer weights).
if allinteger:
verifyOptimum()
return mate
