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The Nature of Computation$
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Cristopher Moore and Stephan Mertens

Print publication date: 2011

Print ISBN-13: 9780199233212

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199233212.001.0001

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Needles in a Haystack: the Class NP

Needles in a Haystack: the Class NP

(p.94) (p.95) Chapter 4 Needles in a Haystack: the Class NP
The Nature of Computation

Cristopher Moore

Stephan Mertens

Oxford University Press

NP refers to a class of decision problems in which yes-instances are easy to verify. That is: a decision problem is in NP if, whenever the answer for a particular instance is ‘yes’, there is a simple proof of this fact. Given a graph, a Hamiltonian path is a path that visits each vertex exactly once. This chapter examines the NP class of problems, first by considering the proverbial task of finding a needle in a haystack. It then looks at several definitions of NP and discusses proofs, witnesses, and lucky guesses. It shows that NP encompasses a wide range of fundamental problems, from coloring maps to untying knots and satisfying systems of constraints. It describes some of the classic problems in NP and highlights some reductions between these problems by transforming questions about graphs into questions about Boolean formulas or vice versa. The chapter also analyzes primality in NP.

Keywords:   decision problems, NP, Hamiltonian path, proofs, witnesses, lucky guesses, knots, graphs, Boolean formulas, primality

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