def test_trivial5(self):
"""Path"""
G = nx.Graph()
G.add_edge(1, 2, weight=5)
G.add_edge(2, 3, weight=11)
G.add_edge(3, 4, weight=5)
assert edges_equal(nx.max_weight_matching(G),
matching_dict_to_set({
2: 3,
3: 2
}))
assert edges_equal(nx.max_weight_matching(G, 1),
matching_dict_to_set({
1: 2,
2: 1,
3: 4,
4: 3
}))
assert edges_equal(nx.min_weight_matching(G),
matching_dict_to_set({
1: 2,
3: 4
}))
assert edges_equal(nx.min_weight_matching(G, 1),
matching_dict_to_set({
1: 2,
3: 4
}))
def test_negative_weights(self):
"""Negative weights"""
G = nx.Graph()
G.add_edge(1, 2, weight=2)
G.add_edge(1, 3, weight=-2)
G.add_edge(2, 3, weight=1)
G.add_edge(2, 4, weight=-1)
G.add_edge(3, 4, weight=-6)
assert edges_equal(nx.max_weight_matching(G),
matching_dict_to_set({
1: 2,
2: 1
}))
assert edges_equal(nx.max_weight_matching(G, 1),
matching_dict_to_set({
1: 3,
2: 4,
3: 1,
4: 2
}))
assert edges_equal(nx.min_weight_matching(G),
matching_dict_to_set({
1: 2,
3: 4
}))
assert edges_equal(nx.min_weight_matching(G, 1),
matching_dict_to_set({
1: 2,
3: 4
}))
def test_s_blossom(self):
"""Create S-blossom and use it for augmentation:"""
G = nx.Graph()
G.add_weighted_edges_from([(1, 2, 8), (1, 3, 9), (2, 3, 10),
(3, 4, 7)])
answer = matching_dict_to_set({1: 2, 2: 1, 3: 4, 4: 3})
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
G.add_weighted_edges_from([(1, 6, 5), (4, 5, 6)])
answer = matching_dict_to_set({1: 6, 2: 3, 3: 2, 4: 5, 5: 4, 6: 1})
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_nested_s_blossom_relabel(self):
"""Create S-blossom, relabel as S, include in nested S-blossom:"""
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 10),
(1, 7, 10),
(2, 3, 12),
(3, 4, 20),
(3, 5, 20),
(4, 5, 25),
(5, 6, 10),
(6, 7, 10),
(7, 8, 8),
])
answer = matching_dict_to_set({
1: 2,
2: 1,
3: 4,
4: 3,
5: 6,
6: 5,
7: 8,
8: 7
})
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_nested_s_blossom_relabel_expand(self):
"""Create nested S-blossom, relabel as T, expand:"""
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 19),
(1, 3, 20),
(1, 8, 8),
(2, 3, 25),
(2, 4, 18),
(3, 5, 18),
(4, 5, 13),
(4, 7, 7),
(5, 6, 7),
])
answer = matching_dict_to_set({
1: 8,
2: 3,
3: 2,
4: 7,
5: 6,
6: 5,
7: 4,
8: 1
})
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_nasty_blossom1(self):
"""Create blossom, relabel as T in more than one way, expand,
augment:
"""
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 45),
(1, 5, 45),
(2, 3, 50),
(3, 4, 45),
(4, 5, 50),
(1, 6, 30),
(3, 9, 35),
(4, 8, 35),
(5, 7, 26),
(9, 10, 5),
])
ansdict = {
1: 6,
2: 3,
3: 2,
4: 8,
5: 7,
6: 1,
7: 5,
8: 4,
9: 10,
10: 9
}
answer = matching_dict_to_set(ansdict)
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_nested_s_blossom_expand(self):
"""Create nested S-blossom, augment, expand recursively:"""
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 8),
(1, 3, 8),
(2, 3, 10),
(2, 4, 12),
(3, 5, 12),
(4, 5, 14),
(4, 6, 12),
(5, 7, 12),
(6, 7, 14),
(7, 8, 12),
])
answer = matching_dict_to_set({
1: 2,
2: 1,
3: 5,
4: 6,
5: 3,
6: 4,
7: 8,
8: 7
})
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_s_blossom_relabel_expand(self):
"""Create S-blossom, relabel as T, expand:"""
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 23),
(1, 5, 22),
(1, 6, 15),
(2, 3, 25),
(3, 4, 22),
(4, 5, 25),
(4, 8, 14),
(5, 7, 13),
])
answer = matching_dict_to_set({
1: 6,
2: 3,
3: 2,
4: 8,
5: 7,
6: 1,
7: 5,
8: 4
})
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def christofides(G, weight="weight", tree=None):
"""Approximate a solution of the traveling salesman problem
Compute a 3/2-approximation of the traveling salesman problem
in a complete undirected graph using Christofides [1]_ algorithm.
Parameters
----------
G : Graph
`G` should be a complete weighted undirected graph.
The distance between all pairs of nodes should be included.
weight : string, optional (default="weight")
Edge data key corresponding to the edge weight.
If any edge does not have this attribute the weight is set to 1.
tree : NetworkX graph or None (default: None)
A minimum spanning tree of G. Or, if None, the minimum spanning
tree is computed using :func:`networkx.minimum_spanning_tree`
Returns
-------
list
List of nodes in `G` along a cycle with a 3/2-approximation of
the minimal Hamiltonian cycle.
References
----------
.. [1] Christofides, Nicos. "Worst-case analysis of a new heuristic for
the travelling salesman problem." No. RR-388. Carnegie-Mellon Univ
Pittsburgh Pa Management Sciences Research Group, 1976.
"""
# Remove selfloops if necessary
loop_nodes = nx.nodes_with_selfloops(G)
try:
node = next(loop_nodes)
except StopIteration:
pass
else:
G = G.copy()
G.remove_edge(node, node)
G.remove_edges_from((n, n) for n in loop_nodes)
# Check that G is a complete graph
N = len(G) - 1
# This check ignores selfloops which is what we want here.
if any(len(nbrdict) != N for n, nbrdict in G.adj.items()):
raise nx.NetworkXError("G must be a complete graph.")
if tree is None:
tree = nx.minimum_spanning_tree(G, weight=weight)
L = G.copy()
L.remove_nodes_from([v for v, degree in tree.degree if not (degree % 2)])
MG = nx.MultiGraph()
MG.add_edges_from(tree.edges)
edges = nx.min_weight_matching(L, maxcardinality=True, weight=weight)
MG.add_edges_from(edges)
return _shortcutting(nx.eulerian_circuit(MG))
def test_s_t_blossom(self):
"""Create S-blossom, relabel as T-blossom, use for augmentation:"""
G = nx.Graph()
G.add_weighted_edges_from([(1, 2, 9), (1, 3, 8), (2, 3, 10), (1, 4, 5),
(4, 5, 4), (1, 6, 3)])
answer = matching_dict_to_set({1: 6, 2: 3, 3: 2, 4: 5, 5: 4, 6: 1})
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
G.add_edge(4, 5, weight=3)
G.add_edge(1, 6, weight=4)
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
G.remove_edge(1, 6)
G.add_edge(3, 6, weight=4)
answer = matching_dict_to_set({1: 2, 2: 1, 3: 6, 4: 5, 5: 4, 6: 3})
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_trivial3(self):
"""Single edge"""
G = nx.Graph()
G.add_edge(0, 1)
assert edges_equal(nx.max_weight_matching(G),
matching_dict_to_set({
0: 1,
1: 0
}))
assert edges_equal(nx.min_weight_matching(G),
matching_dict_to_set({
0: 1,
1: 0
}))
def test_trivial6(self):
"""Small graph with arbitrary weight attribute"""
G = nx.Graph()
G.add_edge("one", "two", weight=10, abcd=11)
G.add_edge("two", "three", weight=11, abcd=10)
assert edges_equal(
nx.max_weight_matching(G, weight="abcd"),
matching_dict_to_set({
"one": "two",
"two": "one"
}),
)
assert edges_equal(
nx.min_weight_matching(G, weight="abcd"),
matching_dict_to_set({"three": "two"}),
)
def test_trivial4(self):
"""Small graph"""
G = nx.Graph()
G.add_edge("one", "two", weight=10)
G.add_edge("two", "three", weight=11)
assert edges_equal(
nx.max_weight_matching(G),
matching_dict_to_set({
"three": "two",
"two": "three"
}),
)
assert edges_equal(
nx.min_weight_matching(G),
matching_dict_to_set({
"one": "two",
"two": "one"
}),
)
def test_nasty_blossom2(self):
"""Again but slightly different:"""
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 45),
(1, 5, 45),
(2, 3, 50),
(3, 4, 45),
(4, 5, 50),
(1, 6, 30),
(3, 9, 35),
(4, 8, 26),
(5, 7, 40),
(9, 10, 5),
])
ans = {1: 6, 2: 3, 3: 2, 4: 8, 5: 7, 6: 1, 7: 5, 8: 4, 9: 10, 10: 9}
answer = matching_dict_to_set(ans)
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_nested_s_blossom(self):
"""Create nested S-blossom, use for augmentation:"""
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 9),
(1, 3, 9),
(2, 3, 10),
(2, 4, 8),
(3, 5, 8),
(4, 5, 10),
(5, 6, 6),
])
dict_format = {1: 3, 2: 4, 3: 1, 4: 2, 5: 6, 6: 5}
expected = {frozenset(e) for e in matching_dict_to_set(dict_format)}
answer = {frozenset(e) for e in nx.max_weight_matching(G)}
assert answer == expected
answer = {frozenset(e) for e in nx.min_weight_matching(G)}
assert answer == expected
def test_nasty_blossom_expand_recursively(self):
"""Create nested S-blossom, relabel as S, expand recursively:"""
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 40),
(1, 3, 40),
(2, 3, 60),
(2, 4, 55),
(3, 5, 55),
(4, 5, 50),
(1, 8, 15),
(5, 7, 30),
(7, 6, 10),
(8, 10, 10),
(4, 9, 30),
])
ans = {1: 2, 2: 1, 3: 5, 4: 9, 5: 3, 6: 7, 7: 6, 8: 10, 9: 4, 10: 8}
answer = matching_dict_to_set(ans)
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_floating_point_weights(self):
"""Floating point weights"""
G = nx.Graph()
G.add_edge(1, 2, weight=math.pi)
G.add_edge(2, 3, weight=math.exp(1))
G.add_edge(1, 3, weight=3.0)
G.add_edge(1, 4, weight=math.sqrt(2.0))
assert edges_equal(nx.max_weight_matching(G),
matching_dict_to_set({
1: 4,
2: 3,
3: 2,
4: 1
}))
assert edges_equal(nx.min_weight_matching(G),
matching_dict_to_set({
1: 4,
2: 3,
3: 2,
4: 1
}))
def test_nasty_blossom_least_slack(self):
"""Create blossom, relabel as T, expand such that a new
least-slack S-to-free dge is produced, augment:
"""
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 45),
(1, 5, 45),
(2, 3, 50),
(3, 4, 45),
(4, 5, 50),
(1, 6, 30),
(3, 9, 35),
(4, 8, 28),
(5, 7, 26),
(9, 10, 5),
])
ans = {1: 6, 2: 3, 3: 2, 4: 8, 5: 7, 6: 1, 7: 5, 8: 4, 9: 10, 10: 9}
answer = matching_dict_to_set(ans)
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_nasty_blossom_augmenting(self):
"""Create nested blossom, relabel as T in more than one way"""
# expand outer blossom such that inner blossom ends up on an
# augmenting path:
G = nx.Graph()
G.add_weighted_edges_from([
(1, 2, 45),
(1, 7, 45),
(2, 3, 50),
(3, 4, 45),
(4, 5, 95),
(4, 6, 94),
(5, 6, 94),
(6, 7, 50),
(1, 8, 30),
(3, 11, 35),
(5, 9, 36),
(7, 10, 26),
(11, 12, 5),
])
ans = {
1: 8,
2: 3,
3: 2,
4: 6,
5: 9,
6: 4,
7: 10,
8: 1,
9: 5,
10: 7,
11: 12,
12: 11,
}
answer = matching_dict_to_set(ans)
assert edges_equal(nx.max_weight_matching(G), answer)
assert edges_equal(nx.min_weight_matching(G), answer)
def test_trivial1(self):
"""Empty graph"""
G = nx.Graph()
assert nx.max_weight_matching(G) == set()
assert nx.min_weight_matching(G) == set()
def test_trivial2(self):
"""Self loop"""
G = nx.Graph()
G.add_edge(0, 0, weight=100)
assert nx.max_weight_matching(G) == set()
assert nx.min_weight_matching(G) == set()
