CS 312: Algorithm Analysis

Lecture #28: Network Flow and Cuts

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Slides by: Eric Ringger, with contributions from Mike Jones

Objectives

Understand the idea of “changing the problem” in the max flow algorithm

Solve min-cut (using max-flow) Solve Bipartite Matching (using max-flow) See example of Padded matrix-vector

representation

Volunteer

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Max Flow

Key points When we transform the problem we create

“imaginary edges” However, in the end our selected paths

always produce a flow schedule that involves only the original edges.

Cut

An (s,t)-cut partitions the vertices into two disjoint sets:L contains sR contains t

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To compute the capacity of a cut, add up the capacitiesof edges that flow across the cut from L to R (not R to L!).

L:SourceSide

R:SinkSide

Max-flow min-cut Theorem

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Max flow is less than or equal to the capacity of any cut. Max flow is equal to the capacity of the min cut.

Another Example

We can now find the min cut in the original graph by finding all of the nodes that are reachable from the source in the final residual graph . The min cut value is equal to the max flow.

Bipartite Matching

• We can use our solution to the max-flow problem to find maximal matchings.

Big ideas Max-flow is a nice graphical version of the Simplex

algorithm Proceed greedily, but use slack variables so that you can

change your decisions later on. Find a quick & dirty solution (not necessarily optimal) in order to

remove some slack from the problem; then transform the problem and try again!

Max-flow can be used to solve other interesting problems Min-cut Matching

LP: Padded Matrices & VectorsStandard Form (Step #1)

Subject to:

Matrix/vector notation (Step #1)

Subject to:

Where is an optional constant in the objective function

Standard Form (Step #2)

Subject to:

Matrix/vector notation (Step #2)

Subject to:

Where are the indices of the nonbasic (original) variables and are the indices of the basic (slack) variables.

Project #6 Linear Programming Important: LP Notes available on the schedule

Works through the ideas up through the Simplex algorithm Sets you up to succeed on project #6 Includes the pseudo-code for Simplex (and Pivot)

HW #21: Formulate problem for Proj. #6 NO 2-hour time limit

Early day: next Thursday Due: next Friday How to Test: Use online solver to verify your result!