Maximum Flows

1
Maximum Flows

• Lecture 4 Jan 19

2
Network transmission

• Given a directed graph G
• A source node s
• A sink node t
• Goal To send as much information from s to t

3
Maximum flow
receiver
Capacity constraint (bits per second)
4
Flows

• An s-t flow is a function f which satisfies
• (capacity constraint)
• (conservation of flows)

• An s-t flow is a function f which satisfies
• (capacity constraint)
• (conservation of flows)

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Value of the flow

Maximum flow problem maximize this value
3
4
G
9
10
7
8
6
6
10
10
2
0
9
10
9
s
t
10
10
9
Value 19
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An upper bound
receiver
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Cuts

• An s-t cut is a set of edges whose removal
  disconnect s and t
• The capacity of a cut is defined as the sum of
  the capacity of the edges in the cut

Minimum s-t cut problem minimize this capacity
of a s-t cut
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Flows cuts

• Let C be a cut and S be the connected component
  of G-C containing s. Then

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Reverse?

• Value of max s-t flow capacity of min s-t cut
• Suppose every s-t cut has capacity at least 4, is
  it true that there is a max s-t flow of value 2?

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Main result

• (Ford Fulkerson 1956)
• Max flow Min cut
• A polynomial time algorithm

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Greedy method?

• Find an s-t path where every edge has f(e) lt c(e)
• Add this path to the flow
• Repeat until no such path can be found.
• Does it work?

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A counterexample

• How to proceed?

Hint Find an augmenting path
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Residual graph

• Key idea allow flows to push back

f(e) 2
c(e) 10
c(e) 8
c(e) 2
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Finding an augmenting path

• Find an s-t path in the residual graph
• Add it to the current flow to obtain a larger
  flow. Why?

1. Flow conservations
2. More flow going out from s

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Ford-Fulkerson Algorithm

• Start from an empty flow f
• While there is an s-t path in G
  G G
  P, and update f
• Return f

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Max-flow min-cut theorem

• Consider the set S of all vertices reachable from
  s
• So, s is in S, but t is not in S
• No incoming flow coming in S
• Achieve full capacity from S to T

Min cut!
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Complexity

• Assume edge capacity between 1 to C
• At most nC iterations
• Finding an s-t path can be done in O(m) time
• Total running time O(nmC)

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Speeding up

• Find a shortest s-t path ? time
• Capacity scaling

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Integrality theorem

• If every edge has integer capacity,
• then there is a flow of integer value.

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Applications

• of the algorithm
• of the min-max theorem
• of the integrality theorem

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Multi-source multi-sink

• Contrast with multicommodity flow which is
  NP-hard.

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Bipartite matching and more

• Bipartite matching of size k ? max flow of size k
• Work even for d-matching

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Edge disjoint paths

• Find the maximum number of edge disjoint s-t paths

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Graph connectivity

• Minimum number of edges to disconnect a graph
• Menger 1927 Max number of edge-disjoint s-t
  paths
• Min size of an s-t cut.

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Project selection

• Precedence constraints
• Each job has a revenus (can be negative)
• Goal Choose a feasible subset to maximize
  revenue.
• Idea minimize the jobs that lost money,
• and maximize the jobs that gain money.

26
Leaque winner

• See if your favorite team can still win the leaque