Table of Contents
CSE 589 -- Lecture 3
Plan for Today
Maximum Flow
Max-flow outline:
Properties of Flow: f(v,w) -- flow on edge (v,w)
An augmenting path with respect to a given flow f is a
Using an augmenting path to increase flow
Augmenting Path Theorem: A flow f is maximum iff it admits no augmenting path
=> Celebrated Max-flow Min-Cut Theorem
Residual Graph w.r.t. flow f
Ford-Fulkerson Method (G,s,t)
Edmonds-Karp
The shortest path distance from v to t in Rf is non-decreasing.
Shortest paths non-decreasing, cont.
Lemma: between any two consecutive saturations of (v,w), both d(v) and d(w) increase by at least 2.
=> Running time of Edmonds-Karp is O(m2n)
Fastest max-flow algorithms: preflow-push
Preflow-push algorithms
Some applications of max-flow and max-flow min-cut theorem
Bipartite Matching
Network Connectivity
Video on Demand
Other network flow problems: 1. With lower bounds on flow.
Determining feasibility
Other network flow problems: 2. Minimum flow
Shipping Problem
Other network flow problems: 3. Min-cost max-flow
Classical application: Transportation Problem
Example: Fiat makes Uno and Ferrari’s.
Disk head scheduling
Set it up as min-cost max-flow.
New Topic: Linear Programming
Big Bucks!
And more...
An example: The diet problem
Visually… x= peanut butter, y = steak
Optimal vector occurs at corner of feasible set!
General Form of a Linear Program.
The Feasible Set
The Simplex Method
Application: Optimal Pipeline
Simplex Algorithm: An Example
Example of Simplex Method, continued.
Example of Simplex Method, continued.
Example of Simplex Method, continued.
Example of Simplex Method, continued.
What were we doing?
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