Table of Contents
CSE 589 -- Lecture 4
Announcements
Plan for Today
Linear Programming
Feasible Set
An example: The diet problem
Visually… x= peanut butter, y = steak
Optimal vector occurs at corner of feasible set!
Optimal vector occurs at corner of feasible set!
The Feasible Set
The Simplex Method
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?
The Simplex Algorithm and beyond….
A Central Result of LP Theory Duality Theorem
Primal: Minimize cx subject to Ax >= b, x>=0 Dual: Maximize yb subject to yA <=c, y>=0
Simple Example
More general diet problem
Example
What’s going on?
Duality Theorem
For [x(st), x(pb)] and [y(s-prot), y(s-carbo)] feasible =>
Practical Use of Duality
Application: Optimal Pipeline
Summary of Linear Programming
New Topic: Easy vs. Hard Problems
Hardest problems in NP are NP-complete
Most notorious hard graph problem
Clique
Independent Set
Hamiltonian Cycle
Graph Partition
Vertex Coloring
Steiner Tree
Integer Linear Programming
Satisfiability
How you prove a problem P is NP-complete.
Example: Showing Independent Set is NP-complete.
Your turn! Prove that TSP is NP-complete.
Solving NP-complete Problems
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