二分图所有可能的最大匹配
All possible maximum matchings of a bipartite graph
我正在使用 networkx to find the maximum cardinality matching 二分图。
特定图形的匹配边不是唯一的。
有没有办法找到所有的最大匹配？
对于下面的例子，下面的所有边都可以是最大匹配：
{1: 2, 2: 1} 或 {1: 3, 3: 1} 或 {1: 4, 4: 1}
import networkx as nx
import matplotlib.pyplot as plt
G = nx.MultiDiGraph()
edges = [(1,3), (1,4), (1,2)]
nx.is_bipartite(G)
True
nx.draw(G, with_labels=True)
plt.show()
不幸的是，
nx.bipartite.maximum_matching(G)
仅returns
{1: 2, 2: 1}
有没有办法让我也得到其他组合？
Takeaki Uno 的论文 "Algorithms for Enumerating All Perfect, Maximum and Maximal Matchings in Bipartite Graphs" 对此有一个算法。 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.107.8179&rep=rep1&type=pdf
定理 2 说
“二分图中的最大匹配可以在 O(mn^1/2+
nNm) 时间和 O(m) space，其中 Nm 是 G 中的最大匹配数。"
我读过 Uno 的作品并试图想出一个实现。下面是我非常冗长的代码和一个工作示例。在这种特殊情况下，有 4 个 "feasible" 个顶点（根据 Uno 的术语），所以用一个已经覆盖的顶点切换每个顶点，你总共有 2^4 = 16 个不同的可能最大匹配。
我不得不承认我是图论的新手，我并没有完全遵循 Uno 的流程，其中存在细微差别，而且大多数情况下我没有尝试进行任何优化。我确实很难理解这篇论文，因为我认为解释不是很完美，而且数字中可能有错误。所以请谨慎使用，如果您能帮助优化它，那将是非常棒的！
import networkx as nx
from networkx import bipartite
def plotGraph(graph):
import matplotlib.pyplot as plt
fig=plt.figure()
ax=fig.add_subplot(111)
pos=[(ii[1],ii[0]) for ii in graph.nodes()]
pos_dict=dict(zip(graph.nodes(),pos))
nx.draw(graph,pos=pos_dict,ax=ax,with_labels=True)
plt.show(block=False)
return
def formDirected(g,match):
'''Form directed graph D from G and matching M.
<g>: undirected bipartite graph. Nodes are separated by their
'bipartite' attribute.
<match>: list of edges forming a matching of <g>.
Return <d>: directed graph, with edges in <match> pointing from set-0
(bipartite attribute ==0) to set-1 (bipartite attrbiute==1),
and the other edges in <g> but not in <matching> pointing
from set-1 to set-0.
'''
d=nx.DiGraph()
for ee in g.edges():
if ee in match or (ee[1],ee[0]) in match:
if g.node[ee[0]]['bipartite']==0:
d.add_edge(ee[0],ee[1])
else:
d.add_edge(ee[1],ee[0])
else:
if g.node[ee[0]]['bipartite']==0:
d.add_edge(ee[1],ee[0])
else:
d.add_edge(ee[0],ee[1])
return d
def enumMaximumMatching(g):
'''Find all maximum matchings in an undirected bipartite graph.
<g>: undirected bipartite graph. Nodes are separated by their
'bipartite' attribute.
Return <all_matches>: list, each is a list of edges forming a maximum
matching of <g>.
'''
all_matches=[]
#----------------Find one matching M----------------
match=bipartite.hopcroft_karp_matching(g)
#---------------Re-orient match arcs---------------
match2=[]
for kk,vv in match.items():
if g.node[kk]['bipartite']==0:
match2.append((kk,vv))
match=match2
all_matches.append(match)
#-----------------Enter recursion-----------------
all_matches=enumMaximumMatchingIter(g,match,all_matches,None)
return all_matches
def enumMaximumMatchingIter(g,match,all_matches,add_e=None):
'''Recurively search maximum matchings.
<g>: undirected bipartite graph. Nodes are separated by their
'bipartite' attribute.
<match>: list of edges forming one maximum matching of <g>.
<all_matches>: list, each is a list of edges forming a maximum
matching of <g>. Newly found matchings will be appended
into this list.
<add_e>: tuple, the edge used to form subproblems. If not None,
will be added to each newly found matchings.
Return <all_matches>: updated list of all maximum matchings.
'''
#---------------Form directed graph D---------------
d=formDirected(g,match)
#-----------------Find cycles in D-----------------
cycles=list(nx.simple_cycles(d))
if len(cycles)==0:
#---------If no cycle, find a feasible path---------
all_uncovered=set(g.node).difference(set([ii[0] for ii in match]))
all_uncovered=all_uncovered.difference(set([ii[1] for ii in match]))
all_uncovered=list(all_uncovered)
#--------------If no path, terminiate--------------
if len(all_uncovered)==0:
return all_matches
#----------Find a length 2 feasible path----------
idx=0
uncovered=all_uncovered[idx]
while True:
if uncovered not in nx.isolates(g):
paths=nx.single_source_shortest_path(d,uncovered,cutoff=2)
len2paths=[vv for kk,vv in paths.items() if len(vv)==3]
if len(len2paths)>0:
reversed=False
break
#----------------Try reversed path----------------
paths_rev=nx.single_source_shortest_path(d.reverse(),uncovered,cutoff=2)
len2paths=[vv for kk,vv in paths_rev.items() if len(vv)==3]
if len(len2paths)>0:
reversed=True
break
idx+=1
if idx>len(all_uncovered)-1:
return all_matches
uncovered=all_uncovered[idx]
#-------------Create a new matching M'-------------
len2path=len2paths[0]
if reversed:
len2path=len2path[::-1]
len2path=zip(len2path[:-1],len2path[1:])
new_match=[]
for ee in d.edges():
if ee in len2path:
if g.node[ee[1]]['bipartite']==0:
new_match.append((ee[1],ee[0]))
else:
if g.node[ee[0]]['bipartite']==0:
new_match.append(ee)
if add_e is not None:
for ii in add_e:
new_match.append(ii)
all_matches.append(new_match)
#---------------------Select e---------------------
e=set(len2path).difference(set(match))
e=list(e)[0]
#-----------------Form subproblems-----------------
g_plus=g.copy()
g_minus=g.copy()
g_plus.remove_node(e[0])
g_plus.remove_node(e[1])
g_minus.remove_edge(e[0],e[1])
add_e_new=[e,]
if add_e is not None:
add_e_new.extend(add_e)
all_matches=enumMaximumMatchingIter(g_minus,match,all_matches,add_e)
all_matches=enumMaximumMatchingIter(g_plus,new_match,all_matches,add_e_new)
else:
#----------------Find a cycle in D----------------
cycle=cycles[0]
cycle.append(cycle[0])
cycle=zip(cycle[:-1],cycle[1:])
#-------------Create a new matching M'-------------
new_match=[]
for ee in d.edges():
if ee in cycle:
if g.node[ee[1]]['bipartite']==0:
new_match.append((ee[1],ee[0]))
else:
if g.node[ee[0]]['bipartite']==0:
new_match.append(ee)
if add_e is not None:
for ii in add_e:
new_match.append(ii)
all_matches.append(new_match)
#-----------------Choose an edge E-----------------
e=set(match).intersection(set(cycle))
e=list(e)[0]
#-----------------Form subproblems-----------------
g_plus=g.copy()
g_minus=g.copy()
g_plus.remove_node(e[0])
g_plus.remove_node(e[1])
g_minus.remove_edge(e[0],e[1])
add_e_new=[e,]
if add_e is not None:
add_e_new.extend(add_e)
all_matches=enumMaximumMatchingIter(g_plus,match,all_matches,add_e_new)
all_matches=enumMaximumMatchingIter(g_minus,new_match,all_matches,add_e)
return all_matches
if __name__=='__main__':
g=nx.Graph()
edges=[
[(1,0), (0,0)],
[(1,1), (0,0)],
[(1,2), (0,2)],
[(1,3), (0,2)],
[(1,4), (0,3)],
[(1,4), (0,5)],
[(1,5), (0,2)],
[(1,5), (0,4)],
[(1,6), (0,1)],
[(1,6), (0,4)],
[(1,6), (0,6)]
]
for ii in edges:
g.add_node(ii[0],bipartite=0)
g.add_node(ii[1],bipartite=1)
g.add_edges_from(edges)
plotGraph(g)
all_matches=enumMaximumMatching(g)
for mm in all_matches:
g_match=nx.Graph()
for ii in mm:
g_match.add_edge(ii[0],ii[1])
plotGraph(g_match)
我正在使用 networkx to find the maximum cardinality matching 二分图。
特定图形的匹配边不是唯一的。
有没有办法找到所有的最大匹配？
对于下面的例子，下面的所有边都可以是最大匹配：
{1: 2, 2: 1} 或 {1: 3, 3: 1} 或 {1: 4, 4: 1}
import networkx as nx
import matplotlib.pyplot as plt
G = nx.MultiDiGraph()
edges = [(1,3), (1,4), (1,2)]
nx.is_bipartite(G)
True
nx.draw(G, with_labels=True)
plt.show()
不幸的是，
nx.bipartite.maximum_matching(G)
仅returns
{1: 2, 2: 1}
有没有办法让我也得到其他组合？
Takeaki Uno 的论文 "Algorithms for Enumerating All Perfect, Maximum and Maximal Matchings in Bipartite Graphs" 对此有一个算法。 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.107.8179&rep=rep1&type=pdf
定理 2 说 “二分图中的最大匹配可以在 O(mn^1/2+ nNm) 时间和 O(m) space，其中 Nm 是 G 中的最大匹配数。"
我读过 Uno 的作品并试图想出一个实现。下面是我非常冗长的代码和一个工作示例。在这种特殊情况下，有 4 个 "feasible" 个顶点（根据 Uno 的术语），所以用一个已经覆盖的顶点切换每个顶点，你总共有 2^4 = 16 个不同的可能最大匹配。
我不得不承认我是图论的新手，我并没有完全遵循 Uno 的流程，其中存在细微差别，而且大多数情况下我没有尝试进行任何优化。我确实很难理解这篇论文，因为我认为解释不是很完美，而且数字中可能有错误。所以请谨慎使用，如果您能帮助优化它，那将是非常棒的！
import networkx as nx
from networkx import bipartite
def plotGraph(graph):
import matplotlib.pyplot as plt
fig=plt.figure()
ax=fig.add_subplot(111)
pos=[(ii[1],ii[0]) for ii in graph.nodes()]
pos_dict=dict(zip(graph.nodes(),pos))
nx.draw(graph,pos=pos_dict,ax=ax,with_labels=True)
plt.show(block=False)
return
def formDirected(g,match):
'''Form directed graph D from G and matching M.
<g>: undirected bipartite graph. Nodes are separated by their
'bipartite' attribute.
<match>: list of edges forming a matching of <g>.
Return <d>: directed graph, with edges in <match> pointing from set-0
(bipartite attribute ==0) to set-1 (bipartite attrbiute==1),
and the other edges in <g> but not in <matching> pointing
from set-1 to set-0.
'''
d=nx.DiGraph()
for ee in g.edges():
if ee in match or (ee[1],ee[0]) in match:
if g.node[ee[0]]['bipartite']==0:
d.add_edge(ee[0],ee[1])
else:
d.add_edge(ee[1],ee[0])
else:
if g.node[ee[0]]['bipartite']==0:
d.add_edge(ee[1],ee[0])
else:
d.add_edge(ee[0],ee[1])
return d
def enumMaximumMatching(g):
'''Find all maximum matchings in an undirected bipartite graph.
<g>: undirected bipartite graph. Nodes are separated by their
'bipartite' attribute.
Return <all_matches>: list, each is a list of edges forming a maximum
matching of <g>.
'''
all_matches=[]
#----------------Find one matching M----------------
match=bipartite.hopcroft_karp_matching(g)
#---------------Re-orient match arcs---------------
match2=[]
for kk,vv in match.items():
if g.node[kk]['bipartite']==0:
match2.append((kk,vv))
match=match2
all_matches.append(match)
#-----------------Enter recursion-----------------
all_matches=enumMaximumMatchingIter(g,match,all_matches,None)
return all_matches
def enumMaximumMatchingIter(g,match,all_matches,add_e=None):
'''Recurively search maximum matchings.
<g>: undirected bipartite graph. Nodes are separated by their
'bipartite' attribute.
<match>: list of edges forming one maximum matching of <g>.
<all_matches>: list, each is a list of edges forming a maximum
matching of <g>. Newly found matchings will be appended
into this list.
<add_e>: tuple, the edge used to form subproblems. If not None,
will be added to each newly found matchings.
Return <all_matches>: updated list of all maximum matchings.
'''
#---------------Form directed graph D---------------
d=formDirected(g,match)
#-----------------Find cycles in D-----------------
cycles=list(nx.simple_cycles(d))
if len(cycles)==0:
#---------If no cycle, find a feasible path---------
all_uncovered=set(g.node).difference(set([ii[0] for ii in match]))
all_uncovered=all_uncovered.difference(set([ii[1] for ii in match]))
all_uncovered=list(all_uncovered)
#--------------If no path, terminiate--------------
if len(all_uncovered)==0:
return all_matches
#----------Find a length 2 feasible path----------
idx=0
uncovered=all_uncovered[idx]
while True:
if uncovered not in nx.isolates(g):
paths=nx.single_source_shortest_path(d,uncovered,cutoff=2)
len2paths=[vv for kk,vv in paths.items() if len(vv)==3]
if len(len2paths)>0:
reversed=False
break
#----------------Try reversed path----------------
paths_rev=nx.single_source_shortest_path(d.reverse(),uncovered,cutoff=2)
len2paths=[vv for kk,vv in paths_rev.items() if len(vv)==3]
if len(len2paths)>0:
reversed=True
break
idx+=1
if idx>len(all_uncovered)-1:
return all_matches
uncovered=all_uncovered[idx]
#-------------Create a new matching M'-------------
len2path=len2paths[0]
if reversed:
len2path=len2path[::-1]
len2path=zip(len2path[:-1],len2path[1:])
new_match=[]
for ee in d.edges():
if ee in len2path:
if g.node[ee[1]]['bipartite']==0:
new_match.append((ee[1],ee[0]))
else:
if g.node[ee[0]]['bipartite']==0:
new_match.append(ee)
if add_e is not None:
for ii in add_e:
new_match.append(ii)
all_matches.append(new_match)
#---------------------Select e---------------------
e=set(len2path).difference(set(match))
e=list(e)[0]
#-----------------Form subproblems-----------------
g_plus=g.copy()
g_minus=g.copy()
g_plus.remove_node(e[0])
g_plus.remove_node(e[1])
g_minus.remove_edge(e[0],e[1])
add_e_new=[e,]
if add_e is not None:
add_e_new.extend(add_e)
all_matches=enumMaximumMatchingIter(g_minus,match,all_matches,add_e)
all_matches=enumMaximumMatchingIter(g_plus,new_match,all_matches,add_e_new)
else:
#----------------Find a cycle in D----------------
cycle=cycles[0]
cycle.append(cycle[0])
cycle=zip(cycle[:-1],cycle[1:])
#-------------Create a new matching M'-------------
new_match=[]
for ee in d.edges():
if ee in cycle:
if g.node[ee[1]]['bipartite']==0:
new_match.append((ee[1],ee[0]))
else:
if g.node[ee[0]]['bipartite']==0:
new_match.append(ee)
if add_e is not None:
for ii in add_e:
new_match.append(ii)
all_matches.append(new_match)
#-----------------Choose an edge E-----------------
e=set(match).intersection(set(cycle))
e=list(e)[0]
#-----------------Form subproblems-----------------
g_plus=g.copy()
g_minus=g.copy()
g_plus.remove_node(e[0])
g_plus.remove_node(e[1])
g_minus.remove_edge(e[0],e[1])
add_e_new=[e,]
if add_e is not None:
add_e_new.extend(add_e)
all_matches=enumMaximumMatchingIter(g_plus,match,all_matches,add_e_new)
all_matches=enumMaximumMatchingIter(g_minus,new_match,all_matches,add_e)
return all_matches
if __name__=='__main__':
g=nx.Graph()
edges=[
[(1,0), (0,0)],
[(1,1), (0,0)],
[(1,2), (0,2)],
[(1,3), (0,2)],
[(1,4), (0,3)],
[(1,4), (0,5)],
[(1,5), (0,2)],
[(1,5), (0,4)],
[(1,6), (0,1)],
[(1,6), (0,4)],
[(1,6), (0,6)]
]
for ii in edges:
g.add_node(ii[0],bipartite=0)
g.add_node(ii[1],bipartite=1)
g.add_edges_from(edges)
plotGraph(g)
all_matches=enumMaximumMatching(g)
for mm in all_matches:
g_match=nx.Graph()
for ii in mm:
g_match.add_edge(ii[0],ii[1])
plotGraph(g_match)