This document discusses the max flow problem and algorithms. It introduces flow networks with source and sink nodes, defines the max flow problem as finding the highest value flow, and shows its equivalence to the min cut problem of finding the lowest capacity edge cut. It then describes Ford-Fulkerson's algorithm, which works by finding augmenting paths in the residual graph to incrementally increase the flow until no more augmenting paths exist, at which point the max flow is found.

1. Flow Network Max Flow Algorithm Fundamental Knowledge
Artificial Intelligence
Max Flow Problem
G. Guérard
Department of Nouvelles Energies
Ecole Supérieure d’Ingénieurs Léonard de Vinci
Lecture 4
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2. Flow Network Max Flow Algorithm Fundamental Knowledge
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1 Flow Network
2 Max Flow Algorithm
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3. Flow Network Max Flow Algorithm Fundamental Knowledge
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Various problems can be reducted into MAX FLOW PROBLEM,
for examples:
NETWORK CONNECTIVITY: Edge-disjoint paths, Network
reliability.
MATCHING: Bipartite matching, Maximum matchings.
VERTEX CAPACITIES: Data mining.
SCHEDULING: Airline scheduling, Project selection.
GRAPH CUTS: Image processing.
ASSIGNEMENT: Baseball elimination, Distributed computing.
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4. Flow Network Max Flow Algorithm Fundamental Knowledge
Min cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problem
Flow network
Let G = (V , E) be a directed graph with two distingished
nodes s, the source, and t, the sink. And a capacity c(e) for
each edge e ∈ E.
Min cut problem
Delete a set of edges to disconnect t from s. Found a set to
minimize the sum of edge’s capacity. A CUT is a node partition
(S, T) such that s is in S and t is in T.
MINIMIZE THE SUM OF WEIGHTS OF EDGES LEAVING S.
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5. Flow Network Max Flow Algorithm Fundamental Knowledge
Min cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problemMin cut problem
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6. Flow Network Max Flow Algorithm Fundamental Knowledge
FlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlows
Flow
A FLOW f is an assignment of weights to edges so that. The
flow can’t go over the capacity of an edge. The flow entering in a
vertex is equal to the flow leaving it. An s − t flow is a function
that satisfies: for each e ∈ E, 0 ≤ f (e) ≤ c(e) (capacity); for
each v ∈ V − {s, t}, ∑
e∈Γ+
f (e) = ∑
e∈Γ−
f (e).
Max flow
Find s − t flow of maximum value v(f ). An simple and easy
way to know the max flow is to sum flow out of s or to sum
flow in to t: v(f ) = ∑
e∈Γ+(s)
f (e).
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7. Flow Network Max Flow Algorithm Fundamental Knowledge
FlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlowsFlows
Flow value Lemma
Let f be any flow, and let (A, B) be any cut. Then, the net flow
sent across the cut is equal to the amount leaving
s:v(f ) = ∑
e∈Γ+(A)
f (e) − ∑
e∈Γ−(A)
f (e).
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8. Flow Network Max Flow Algorithm Fundamental Knowledge
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Weak duality
Let f be any flow, and let (A, B) be any cut. Then the value of
the flow is at most the capacity of the cut.
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9. Flow Network Max Flow Algorithm Fundamental Knowledge
Flows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cutsFlows and cuts
Corollary
Let f be any flow, and let (A, B) be any cut. If v(f ) is equal to
the capacity of the cut, then f is a max flow and (A, B) forms a
min cut.
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10. Flow Network Max Flow Algorithm Fundamental Knowledge
Augmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting path
Find a s − t path where each arc has f (e) < u(e) and augment flow
along it by the minimum of u(e) − f (e). A naive algorithm consists
to repeat this process until you get stuck. This method do not give an
optimal value, we need to be able to backtrack.
Residual graph
Let e be an edge from v to w, u(e) its capacity and f (e) the flow
passing by this edge. A residual graph G = (V , E ) represents each
possible evolution of flows in the graph G = (V , E). To construct G
all the arc that have strictly positive flow are duplicate, the new arc
go from the end node to the start node, its capacity is equal to the
flow. The previous has a capacity equal to u(e) − f (e).
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11. Flow Network Max Flow Algorithm Fundamental Knowledge
Augmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting pathAugmenting path
Augmenting path
Find a s − t path in residual graph, it is called augmenting path.
Once the path is selected, increase flow along forward edges,
decrease flow along backward edges.
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12. Flow Network Max Flow Algorithm Fundamental Knowledge
AlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithm
FORD-FULKERSON’s algorithm is a generic (greedy) method for
solving max flow.
While there exists an augmenting path
find augmenting path P
compute bottleneck capacity of P
augment flow along P
Theorem
A flow f is a max flow iff there are no augmenting paths.
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13. Flow Network Max Flow Algorithm Fundamental Knowledge
AlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithmAlgorithm
THOSE THREE SENTENCES ARE EQUIVALENT:
f is a max flow.
There is no augmenting path relative to f .
There exists a cut whose capacity equals the value of f .
SO, THE FOLLOWING PROBLEMS ARE EQUIVALENT:
Find a max flow.
Find a min cut.
Use residual graph to find bottlenecks (backward edges without
respecting forwward edges).
Use residual graph to find a min cut (minimum value set of
backward edges without respecting forward edges).
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14. Flow Network Max Flow Algorithm Fundamental Knowledge
Good Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting PathGood Augmenting Path
Use care when selecting augmenting paths. Design goal is to
choose augmenting paths so that can find augmenting paths
efficiently and in few iterations, not like:
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15. Flow Network Max Flow Algorithm Fundamental Knowledge
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YOU HAVE TO KNOW BEFORE THE TUTORIAL:
1 Flow definition and max flow problem.
2 Min-cut problem and weak duality.
3 Augmenting path definition.
4 Ford-Fulkerson’s algorithm.
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