Table of Contents
CSE 589 -- Lecture 5
Plan for Today
Easy vs. Hard Problems
Polynomial Time Reductions
Essence of NP-completeness
Definitions, etc.
Proving a Problem is NP-complete
Remarkable Theorem of Steve Cook (1971) Proved that there exist NP-complete problems
Idea of Proof
Traveling Salesman Problem
Clique
Vertex Coloring
Integer Linear Programming
Techniques for Dealing with NP-complete Problems
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Backtracking Example: Finding a 3-coloring of a graph
Graph Recursion Tree
Branch-and-Bound
Example: Minimum number of colors in graph coloring
Branch-and-Bound
How might you obtain such a bound?
Bounds for clique problem
Two things you can do with linear program
PPT Slide
Approximation Algorithms
Example 1: Vertex Cover
The Vertex Cover Produced Is At Most Twice The Size Of The Optimal VC
Example 2: Approximation Algorithm for Euclidean Traveling Salesman Problem
Approximation Algorithm For Euclidean TSP
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Better version of algorithm
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This algorithm has provably performance guarantee
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Randomized Rounding
Example: Global Wiring In A Gate Array
Simplifications
Cast as Integer Program
Randomized Rounding
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Local Search Algorithms
Local Search Procedure for TSP
Does Greedy Local Search Lead Eventually To Optimal Tour?
Solution Spaces
Other Types of Local Moves For TSP Used
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Tabu Search
Augment Greedy Local Search on TSP with Ideas From Tabu Search Paradigm
Limit Acceptance of Moves that Worsen Cost of Current Tour
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Simulated Annealing
Notation
Metropolis Algorithm
What’s going on?
Some Details
Your turn to do an NP-completeness proof
Your turn to analyze an approximation algorithm for an NP-complete problem.
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