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定义: 图G(V,E)是指一个二元组(V(G),E(G)),其中:
举例:
V(G)={v1,v2,v3,v4}
E(G)= {e1,e2,e3,e4,e5,e6}
设G(V,E)为有向图,若在每条边e上定义一个非负权c, 则称图G为一个网络,称c为边e的容量函数,记为c(e)。
若在有向图G(V,E)中有两个不同的顶点vs与vt ,
若顶点vs只有出度没有入度,称vs为图G的源,
若顶点vt只有入度没有出度, 称vt为G的汇,
若顶点v 既不是源也不是汇, 称为v中间顶点。
如图,就是从v1到v9怎么流动,在受每一个有向边的流动最大限制下,才是最大流。大学考试的内容一般都是用手算的,这里我们还是用python来解决最大流问题。
from ortools.graph import pywrapgraph
start_nodes = [0, 0, 0, 1, 1, 2, 2, 3, 3]
end_nodes = [1, 2, 3, 2, 4, 3, 4, 2, 4]
capacities = [20, 30, 10, 40, 30, 10, 20, 5, 20]
max_flow = pywrapgraph.SimpleMaxFlow()
for i in range(0, len(start_nodes)):
max_flow.AddArcWithCapacity(start_nodes[i], end_nodes[i], capacities[i])
# Find the maximum flow between node 0 and node 4.
if max_flow.Solve(0, 4) == max_flow.OPTIMAL:
print('Max flow:', max_flow.OptimalFlow())
print('')
print(' Arc Flow / Capacity')
for i in range(max_flow.NumArcs()):
print('%1s -> %1s %3s / %3s' % (
max_flow.Tail(i),
max_flow.Head(i),
max_flow.Flow(i),
max_flow.Capacity(i)))
print('Source side min-cut:', max_flow.GetSourceSideMinCut())
print('Sink side min-cut:', max_flow.GetSinkSideMinCut())
else:
print('There was an issue with the max flow input.')
from ortools.graph import pywrapgraph
#between each pair. For instance, the arc from node 0 to node 1 has acapacity of 15 and a unit cost of 4.
start_nodes = [ 0, 0, 1, 1, 1, 2, 2, 3, 4]
end_nodes = [ 1, 2, 2, 3, 4, 3, 4, 4, 2]
capacities = [15, 8, 20, 4, 10, 15, 4, 20, 5]
unit_costs = [ 4, 4, 2, 2, 6, 1, 3, 2, 3]
# Define an array of supplies at each node.
supplies = [20, 0, 0, -5, -15]
# Instantiate a SimpleMinCostFlow solver.
min_cost_flow = pywrapgraph.SimpleMinCostFlow()
# Add each arc.
for i in range(0, len(start_nodes)):
min_cost_flow.AddArcWithCapacityAndUnitCost(start_nodes[i], end_nodes[i],
capacities[i], unit_costs[i])
# Add node supplies.
for i in range(0, len(supplies)):
min_cost_flow.SetNodeSupply(i, supplies[i])
# Find the minimum cost flow between node 0 and node 4.
if min_cost_flow.Solve() == min_cost_flow.OPTIMAL:
print('Minimum cost:', min_cost_flow.OptimalCost())
print('')
print(' Arc Flow / Capacity Cost')
for i in range(min_cost_flow.NumArcs()):
cost = min_cost_flow.Flow(i) * min_cost_flow.UnitCost(i)
print('%1s -> %1s %3s / %3s %3s' % (
min_cost_flow.Tail(i),
min_cost_flow.Head(i),
min_cost_flow.Flow(i),
min_cost_flow.Capacity(i),
cost))
else:
print('There was an issue with the min cost flow input.')