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+import bz2
+import importlib.resources
+import os
+import pickle
+
+import pytest
+
+import networkx as nx
+
+
+class TestMinCostFlow:
+ def test_simple_digraph(self):
+ G = nx.DiGraph()
+ G.add_node("a", demand=-5)
+ G.add_node("d", demand=5)
+ G.add_edge("a", "b", weight=3, capacity=4)
+ G.add_edge("a", "c", weight=6, capacity=10)
+ G.add_edge("b", "d", weight=1, capacity=9)
+ G.add_edge("c", "d", weight=2, capacity=5)
+ flowCost, H = nx.network_simplex(G)
+ soln = {"a": {"b": 4, "c": 1}, "b": {"d": 4}, "c": {"d": 1}, "d": {}}
+ assert flowCost == 24
+ assert nx.min_cost_flow_cost(G) == 24
+ assert H == soln
+ assert nx.min_cost_flow(G) == soln
+ assert nx.cost_of_flow(G, H) == 24
+
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == 24
+ assert nx.cost_of_flow(G, H) == 24
+ assert H == soln
+
+ def test_negcycle_infcap(self):
+ G = nx.DiGraph()
+ G.add_node("s", demand=-5)
+ G.add_node("t", demand=5)
+ G.add_edge("s", "a", weight=1, capacity=3)
+ G.add_edge("a", "b", weight=3)
+ G.add_edge("c", "a", weight=-6)
+ G.add_edge("b", "d", weight=1)
+ G.add_edge("d", "c", weight=-2)
+ G.add_edge("d", "t", weight=1, capacity=3)
+ pytest.raises(nx.NetworkXUnfeasible, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXUnbounded, nx.capacity_scaling, G)
+
+ def test_sum_demands_not_zero(self):
+ G = nx.DiGraph()
+ G.add_node("s", demand=-5)
+ G.add_node("t", demand=4)
+ G.add_edge("s", "a", weight=1, capacity=3)
+ G.add_edge("a", "b", weight=3)
+ G.add_edge("a", "c", weight=-6)
+ G.add_edge("b", "d", weight=1)
+ G.add_edge("c", "d", weight=-2)
+ G.add_edge("d", "t", weight=1, capacity=3)
+ pytest.raises(nx.NetworkXUnfeasible, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXUnfeasible, nx.capacity_scaling, G)
+
+ def test_no_flow_satisfying_demands(self):
+ G = nx.DiGraph()
+ G.add_node("s", demand=-5)
+ G.add_node("t", demand=5)
+ G.add_edge("s", "a", weight=1, capacity=3)
+ G.add_edge("a", "b", weight=3)
+ G.add_edge("a", "c", weight=-6)
+ G.add_edge("b", "d", weight=1)
+ G.add_edge("c", "d", weight=-2)
+ G.add_edge("d", "t", weight=1, capacity=3)
+ pytest.raises(nx.NetworkXUnfeasible, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXUnfeasible, nx.capacity_scaling, G)
+
+ def test_transshipment(self):
+ G = nx.DiGraph()
+ G.add_node("a", demand=1)
+ G.add_node("b", demand=-2)
+ G.add_node("c", demand=-2)
+ G.add_node("d", demand=3)
+ G.add_node("e", demand=-4)
+ G.add_node("f", demand=-4)
+ G.add_node("g", demand=3)
+ G.add_node("h", demand=2)
+ G.add_node("r", demand=3)
+ G.add_edge("a", "c", weight=3)
+ G.add_edge("r", "a", weight=2)
+ G.add_edge("b", "a", weight=9)
+ G.add_edge("r", "c", weight=0)
+ G.add_edge("b", "r", weight=-6)
+ G.add_edge("c", "d", weight=5)
+ G.add_edge("e", "r", weight=4)
+ G.add_edge("e", "f", weight=3)
+ G.add_edge("h", "b", weight=4)
+ G.add_edge("f", "d", weight=7)
+ G.add_edge("f", "h", weight=12)
+ G.add_edge("g", "d", weight=12)
+ G.add_edge("f", "g", weight=-1)
+ G.add_edge("h", "g", weight=-10)
+ flowCost, H = nx.network_simplex(G)
+ soln = {
+ "a": {"c": 0},
+ "b": {"a": 0, "r": 2},
+ "c": {"d": 3},
+ "d": {},
+ "e": {"r": 3, "f": 1},
+ "f": {"d": 0, "g": 3, "h": 2},
+ "g": {"d": 0},
+ "h": {"b": 0, "g": 0},
+ "r": {"a": 1, "c": 1},
+ }
+ assert flowCost == 41
+ assert nx.min_cost_flow_cost(G) == 41
+ assert H == soln
+ assert nx.min_cost_flow(G) == soln
+ assert nx.cost_of_flow(G, H) == 41
+
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == 41
+ assert nx.cost_of_flow(G, H) == 41
+ assert H == soln
+
+ def test_max_flow_min_cost(self):
+ G = nx.DiGraph()
+ G.add_edge("s", "a", bandwidth=6)
+ G.add_edge("s", "c", bandwidth=10, cost=10)
+ G.add_edge("a", "b", cost=6)
+ G.add_edge("b", "d", bandwidth=8, cost=7)
+ G.add_edge("c", "d", cost=10)
+ G.add_edge("d", "t", bandwidth=5, cost=5)
+ soln = {
+ "s": {"a": 5, "c": 0},
+ "a": {"b": 5},
+ "b": {"d": 5},
+ "c": {"d": 0},
+ "d": {"t": 5},
+ "t": {},
+ }
+ flow = nx.max_flow_min_cost(G, "s", "t", capacity="bandwidth", weight="cost")
+ assert flow == soln
+ assert nx.cost_of_flow(G, flow, weight="cost") == 90
+
+ G.add_edge("t", "s", cost=-100)
+ flowCost, flow = nx.capacity_scaling(G, capacity="bandwidth", weight="cost")
+ G.remove_edge("t", "s")
+ assert flowCost == -410
+ assert flow["t"]["s"] == 5
+ del flow["t"]["s"]
+ assert flow == soln
+ assert nx.cost_of_flow(G, flow, weight="cost") == 90
+
+ def test_digraph1(self):
+ # From Bradley, S. P., Hax, A. C. and Magnanti, T. L. Applied
+ # Mathematical Programming. Addison-Wesley, 1977.
+ G = nx.DiGraph()
+ G.add_node(1, demand=-20)
+ G.add_node(4, demand=5)
+ G.add_node(5, demand=15)
+ G.add_edges_from(
+ [
+ (1, 2, {"capacity": 15, "weight": 4}),
+ (1, 3, {"capacity": 8, "weight": 4}),
+ (2, 3, {"weight": 2}),
+ (2, 4, {"capacity": 4, "weight": 2}),
+ (2, 5, {"capacity": 10, "weight": 6}),
+ (3, 4, {"capacity": 15, "weight": 1}),
+ (3, 5, {"capacity": 5, "weight": 3}),
+ (4, 5, {"weight": 2}),
+ (5, 3, {"capacity": 4, "weight": 1}),
+ ]
+ )
+ flowCost, H = nx.network_simplex(G)
+ soln = {
+ 1: {2: 12, 3: 8},
+ 2: {3: 8, 4: 4, 5: 0},
+ 3: {4: 11, 5: 5},
+ 4: {5: 10},
+ 5: {3: 0},
+ }
+ assert flowCost == 150
+ assert nx.min_cost_flow_cost(G) == 150
+ assert H == soln
+ assert nx.min_cost_flow(G) == soln
+ assert nx.cost_of_flow(G, H) == 150
+
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == 150
+ assert H == soln
+ assert nx.cost_of_flow(G, H) == 150
+
+ def test_digraph2(self):
+ # Example from ticket #430 from mfrasca. Original source:
+ # http://www.cs.princeton.edu/courses/archive/spr03/cs226/lectures/mincost.4up.pdf, slide 11.
+ G = nx.DiGraph()
+ G.add_edge("s", 1, capacity=12)
+ G.add_edge("s", 2, capacity=6)
+ G.add_edge("s", 3, capacity=14)
+ G.add_edge(1, 2, capacity=11, weight=4)
+ G.add_edge(2, 3, capacity=9, weight=6)
+ G.add_edge(1, 4, capacity=5, weight=5)
+ G.add_edge(1, 5, capacity=2, weight=12)
+ G.add_edge(2, 5, capacity=4, weight=4)
+ G.add_edge(2, 6, capacity=2, weight=6)
+ G.add_edge(3, 6, capacity=31, weight=3)
+ G.add_edge(4, 5, capacity=18, weight=4)
+ G.add_edge(5, 6, capacity=9, weight=5)
+ G.add_edge(4, "t", capacity=3)
+ G.add_edge(5, "t", capacity=7)
+ G.add_edge(6, "t", capacity=22)
+ flow = nx.max_flow_min_cost(G, "s", "t")
+ soln = {
+ 1: {2: 6, 4: 5, 5: 1},
+ 2: {3: 6, 5: 4, 6: 2},
+ 3: {6: 20},
+ 4: {5: 2, "t": 3},
+ 5: {6: 0, "t": 7},
+ 6: {"t": 22},
+ "s": {1: 12, 2: 6, 3: 14},
+ "t": {},
+ }
+ assert flow == soln
+
+ G.add_edge("t", "s", weight=-100)
+ flowCost, flow = nx.capacity_scaling(G)
+ G.remove_edge("t", "s")
+ assert flow["t"]["s"] == 32
+ assert flowCost == -3007
+ del flow["t"]["s"]
+ assert flow == soln
+ assert nx.cost_of_flow(G, flow) == 193
+
+ def test_digraph3(self):
+ """Combinatorial Optimization: Algorithms and Complexity,
+ Papadimitriou Steiglitz at page 140 has an example, 7.1, but that
+ admits multiple solutions, so I alter it a bit. From ticket #430
+ by mfrasca."""
+
+ G = nx.DiGraph()
+ G.add_edge("s", "a")
+ G["s"]["a"].update({0: 2, 1: 4})
+ G.add_edge("s", "b")
+ G["s"]["b"].update({0: 2, 1: 1})
+ G.add_edge("a", "b")
+ G["a"]["b"].update({0: 5, 1: 2})
+ G.add_edge("a", "t")
+ G["a"]["t"].update({0: 1, 1: 5})
+ G.add_edge("b", "a")
+ G["b"]["a"].update({0: 1, 1: 3})
+ G.add_edge("b", "t")
+ G["b"]["t"].update({0: 3, 1: 2})
+
+ "PS.ex.7.1: testing main function"
+ sol = nx.max_flow_min_cost(G, "s", "t", capacity=0, weight=1)
+ flow = sum(v for v in sol["s"].values())
+ assert 4 == flow
+ assert 23 == nx.cost_of_flow(G, sol, weight=1)
+ assert sol["s"] == {"a": 2, "b": 2}
+ assert sol["a"] == {"b": 1, "t": 1}
+ assert sol["b"] == {"a": 0, "t": 3}
+ assert sol["t"] == {}
+
+ G.add_edge("t", "s")
+ G["t"]["s"].update({1: -100})
+ flowCost, sol = nx.capacity_scaling(G, capacity=0, weight=1)
+ G.remove_edge("t", "s")
+ flow = sum(v for v in sol["s"].values())
+ assert 4 == flow
+ assert sol["t"]["s"] == 4
+ assert flowCost == -377
+ del sol["t"]["s"]
+ assert sol["s"] == {"a": 2, "b": 2}
+ assert sol["a"] == {"b": 1, "t": 1}
+ assert sol["b"] == {"a": 0, "t": 3}
+ assert sol["t"] == {}
+ assert nx.cost_of_flow(G, sol, weight=1) == 23
+
+ def test_zero_capacity_edges(self):
+ """Address issue raised in ticket #617 by arv."""
+ G = nx.DiGraph()
+ G.add_edges_from(
+ [
+ (1, 2, {"capacity": 1, "weight": 1}),
+ (1, 5, {"capacity": 1, "weight": 1}),
+ (2, 3, {"capacity": 0, "weight": 1}),
+ (2, 5, {"capacity": 1, "weight": 1}),
+ (5, 3, {"capacity": 2, "weight": 1}),
+ (5, 4, {"capacity": 0, "weight": 1}),
+ (3, 4, {"capacity": 2, "weight": 1}),
+ ]
+ )
+ G.nodes[1]["demand"] = -1
+ G.nodes[2]["demand"] = -1
+ G.nodes[4]["demand"] = 2
+
+ flowCost, H = nx.network_simplex(G)
+ soln = {1: {2: 0, 5: 1}, 2: {3: 0, 5: 1}, 3: {4: 2}, 4: {}, 5: {3: 2, 4: 0}}
+ assert flowCost == 6
+ assert nx.min_cost_flow_cost(G) == 6
+ assert H == soln
+ assert nx.min_cost_flow(G) == soln
+ assert nx.cost_of_flow(G, H) == 6
+
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == 6
+ assert H == soln
+ assert nx.cost_of_flow(G, H) == 6
+
+ def test_digon(self):
+ """Check if digons are handled properly. Taken from ticket
+ #618 by arv."""
+ nodes = [(1, {}), (2, {"demand": -4}), (3, {"demand": 4})]
+ edges = [
+ (1, 2, {"capacity": 3, "weight": 600000}),
+ (2, 1, {"capacity": 2, "weight": 0}),
+ (2, 3, {"capacity": 5, "weight": 714285}),
+ (3, 2, {"capacity": 2, "weight": 0}),
+ ]
+ G = nx.DiGraph(edges)
+ G.add_nodes_from(nodes)
+ flowCost, H = nx.network_simplex(G)
+ soln = {1: {2: 0}, 2: {1: 0, 3: 4}, 3: {2: 0}}
+ assert flowCost == 2857140
+ assert nx.min_cost_flow_cost(G) == 2857140
+ assert H == soln
+ assert nx.min_cost_flow(G) == soln
+ assert nx.cost_of_flow(G, H) == 2857140
+
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == 2857140
+ assert H == soln
+ assert nx.cost_of_flow(G, H) == 2857140
+
+ def test_deadend(self):
+ """Check if one-node cycles are handled properly. Taken from ticket
+ #2906 from @sshraven."""
+ G = nx.DiGraph()
+
+ G.add_nodes_from(range(5), demand=0)
+ G.nodes[4]["demand"] = -13
+ G.nodes[3]["demand"] = 13
+
+ G.add_edges_from([(0, 2), (0, 3), (2, 1)], capacity=20, weight=0.1)
+ pytest.raises(nx.NetworkXUnfeasible, nx.min_cost_flow, G)
+
+ def test_infinite_capacity_neg_digon(self):
+ """An infinite capacity negative cost digon results in an unbounded
+ instance."""
+ nodes = [(1, {}), (2, {"demand": -4}), (3, {"demand": 4})]
+ edges = [
+ (1, 2, {"weight": -600}),
+ (2, 1, {"weight": 0}),
+ (2, 3, {"capacity": 5, "weight": 714285}),
+ (3, 2, {"capacity": 2, "weight": 0}),
+ ]
+ G = nx.DiGraph(edges)
+ G.add_nodes_from(nodes)
+ pytest.raises(nx.NetworkXUnbounded, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXUnbounded, nx.capacity_scaling, G)
+
+ def test_finite_capacity_neg_digon(self):
+ """The digon should receive the maximum amount of flow it can handle.
+ Taken from ticket #749 by @chuongdo."""
+ G = nx.DiGraph()
+ G.add_edge("a", "b", capacity=1, weight=-1)
+ G.add_edge("b", "a", capacity=1, weight=-1)
+ min_cost = -2
+ assert nx.min_cost_flow_cost(G) == min_cost
+
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == -2
+ assert H == {"a": {"b": 1}, "b": {"a": 1}}
+ assert nx.cost_of_flow(G, H) == -2
+
+ def test_multidigraph(self):
+ """Multidigraphs are acceptable."""
+ G = nx.MultiDiGraph()
+ G.add_weighted_edges_from([(1, 2, 1), (2, 3, 2)], weight="capacity")
+ flowCost, H = nx.network_simplex(G)
+ assert flowCost == 0
+ assert H == {1: {2: {0: 0}}, 2: {3: {0: 0}}, 3: {}}
+
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == 0
+ assert H == {1: {2: {0: 0}}, 2: {3: {0: 0}}, 3: {}}
+
+ def test_negative_selfloops(self):
+ """Negative selfloops should cause an exception if uncapacitated and
+ always be saturated otherwise.
+ """
+ G = nx.DiGraph()
+ G.add_edge(1, 1, weight=-1)
+ pytest.raises(nx.NetworkXUnbounded, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXUnbounded, nx.capacity_scaling, G)
+ G[1][1]["capacity"] = 2
+ flowCost, H = nx.network_simplex(G)
+ assert flowCost == -2
+ assert H == {1: {1: 2}}
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == -2
+ assert H == {1: {1: 2}}
+
+ G = nx.MultiDiGraph()
+ G.add_edge(1, 1, "x", weight=-1)
+ G.add_edge(1, 1, "y", weight=1)
+ pytest.raises(nx.NetworkXUnbounded, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXUnbounded, nx.capacity_scaling, G)
+ G[1][1]["x"]["capacity"] = 2
+ flowCost, H = nx.network_simplex(G)
+ assert flowCost == -2
+ assert H == {1: {1: {"x": 2, "y": 0}}}
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == -2
+ assert H == {1: {1: {"x": 2, "y": 0}}}
+
+ def test_bone_shaped(self):
+ # From #1283
+ G = nx.DiGraph()
+ G.add_node(0, demand=-4)
+ G.add_node(1, demand=2)
+ G.add_node(2, demand=2)
+ G.add_node(3, demand=4)
+ G.add_node(4, demand=-2)
+ G.add_node(5, demand=-2)
+ G.add_edge(0, 1, capacity=4)
+ G.add_edge(0, 2, capacity=4)
+ G.add_edge(4, 3, capacity=4)
+ G.add_edge(5, 3, capacity=4)
+ G.add_edge(0, 3, capacity=0)
+ flowCost, H = nx.network_simplex(G)
+ assert flowCost == 0
+ assert H == {0: {1: 2, 2: 2, 3: 0}, 1: {}, 2: {}, 3: {}, 4: {3: 2}, 5: {3: 2}}
+ flowCost, H = nx.capacity_scaling(G)
+ assert flowCost == 0
+ assert H == {0: {1: 2, 2: 2, 3: 0}, 1: {}, 2: {}, 3: {}, 4: {3: 2}, 5: {3: 2}}
+
+ def test_exceptions(self):
+ G = nx.Graph()
+ pytest.raises(nx.NetworkXNotImplemented, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXNotImplemented, nx.capacity_scaling, G)
+ G = nx.MultiGraph()
+ pytest.raises(nx.NetworkXNotImplemented, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXNotImplemented, nx.capacity_scaling, G)
+ G = nx.DiGraph()
+ pytest.raises(nx.NetworkXError, nx.network_simplex, G)
+ # pytest.raises(nx.NetworkXError, nx.capacity_scaling, G)
+ G.add_node(0, demand=float("inf"))
+ pytest.raises(nx.NetworkXError, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXUnfeasible, nx.capacity_scaling, G)
+ G.nodes[0]["demand"] = 0
+ G.add_node(1, demand=0)
+ G.add_edge(0, 1, weight=-float("inf"))
+ pytest.raises(nx.NetworkXError, nx.network_simplex, G)
+ pytest.raises(nx.NetworkXUnfeasible, nx.capacity_scaling, G)
+ G[0][1]["weight"] = 0
+ G.add_edge(0, 0, weight=float("inf"))
+ pytest.raises(nx.NetworkXError, nx.network_simplex, G)
+ # pytest.raises(nx.NetworkXError, nx.capacity_scaling, G)
+ G[0][0]["weight"] = 0
+ G[0][1]["capacity"] = -1
+ pytest.raises(nx.NetworkXUnfeasible, nx.network_simplex, G)
+ # pytest.raises(nx.NetworkXUnfeasible, nx.capacity_scaling, G)
+ G[0][1]["capacity"] = 0
+ G[0][0]["capacity"] = -1
+ pytest.raises(nx.NetworkXUnfeasible, nx.network_simplex, G)
+ # pytest.raises(nx.NetworkXUnfeasible, nx.capacity_scaling, G)
+
+ def test_large(self):
+ fname = (
+ importlib.resources.files("networkx.algorithms.flow.tests")
+ / "netgen-2.gpickle.bz2"
+ )
+ with bz2.BZ2File(fname, "rb") as f:
+ G = pickle.load(f)
+ flowCost, flowDict = nx.network_simplex(G)
+ assert 6749969302 == flowCost
+ assert 6749969302 == nx.cost_of_flow(G, flowDict)
+ flowCost, flowDict = nx.capacity_scaling(G)
+ assert 6749969302 == flowCost
+ assert 6749969302 == nx.cost_of_flow(G, flowDict)
generated by cgit v1.2.3 (git 2.47.1) at 2025-07-27 07:57:43 +0000