Maximum Flows

Lecture 4: Jan 19

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

Maximum flow

sender

receiver

Capacity constraint(bits per second)

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)

Value of the flow

s t 10

10

9

8

4

10

10 6 2

10

3

9

9 9 10

7

0

G:6

Value = 19

Maximum flow problem: maximize this value

An upper bound

sender

receiver

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

Flows ≤ cuts

Let C be a cut and S be the connected component of

G-C containing s. Then:

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?

Main result

(Ford Fulkerson 1956)

Max flow = Min cut

A polynomial time algorithm

Greedy method?

Find an s-t path where every edge has f(e) < c(e)

Add this path to the flow

Repeat until no such path can be found.

Does it work?

A counterexample

How to proceed? Hint: Find an augmenting path

Residual graph

Key idea: allow flows to push back

c(e) = 10

f(e) = 2

c(e) = 8

c(e) = 2

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

Ford-Fulkerson Algorithm

1. Start from an empty flow f

2. While there is an s-t path in G

G := G + P, and update

f

3. Return f

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!

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)

Speeding up

Find a shortest s-t path time

Capacity scaling

Integrality theorem

If every edge has integer capacity,

then there is a flow of integer value.

Applications

of the algorithm

of the min-max theorem

of the integrality theorem

Multi-source multi-sink

Contrast with multicommodity flow which is NP-hard.

Bipartite matching and more

Bipartite matching of size k max flow of size k

Work even for d-matching

Edge disjoint paths

Find the maximum number of edge disjoint s-t paths

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.

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.

Leaque winner

See if your favorite team can still win the leaque