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ECE 595, Section 10Numerical Simulations

Lecture 4:NP

-hardness

Prof. Peter Bermel

January 14, 2013

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Outline

Recap from Friday

Class NP

Non-deterministic Turing machines Reducibility

Cook-Levin theorem

Coping with NP Hardness

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Recap from Friday

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Examples in P

CONNECTED the set of all connected graphs

TRIANGLEFREE the set of all graphs without

a triangle

BIPARTITE the set of all bipartite (distinct)

graphs

TREE the set of all trees

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Class NP Definition

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Examples ofNP

Independent set find a k-size subset of a given graph Gsvertices

Traveling salesman given a set ofn nodes and pairwisedistances, find a closed circuit that visits every node once of

length no more thank

Linear/integer programming given a set of m linearinequalities for n variables, find a set of rational/integersolutions

Graph isomorphism given two nxn adjacency matrices M1

and M2, decide if they define the same graph Composite numbers decide if number N is prime

Factoring for a number N, find a factor M in the interval[L,U]

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Relationship ofP & NP

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Relationship ofP & NP

Key open question: does P=NP?

Some NP problems are in P:

Connectivity breadth-first search

Composite numbers Agrawal, Kayal, and Saxena

solved this a few years ago

Linear programming ellipsoidal algorithm of

Khachiyan

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Relationship ofP & NP

Problems notshown to be in P:

Independent set

Traveling salesman

Subset sum

Integer programming

This general group is known as NP complete:

i.e., not in P unless P=NP

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Non-deterministic Turing machines

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Reducibility

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NP completeness

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Cook-Levin theorem

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Web of reductions

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Coping with NP-hardness

Approximate/heuristic solutions

Example: traveling salesman

Key considerations: Maximum error

Average-case complexity

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Related Complexity Classes

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What ifP=NP?

Any decision (or equivalently, search) problem

could be solved in a reasonable amount of

time

Perfectly optimal designs for all areas of

engineering could be chosen

Automated theory generation for

experimental results

No privacy in the digital domain

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Next Class

Discussion of solving linear algebraicequations

Please read Chapter 2 of NumericalRecipes by W.H. Press et al.

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