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The Golden TicketP, NP, and the Search for the Impossible$
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Lance Fortnow

Print publication date: 2017

Print ISBN-13: 9780691175782

Published to Princeton Scholarship Online: May 2018

DOI: 10.23943/princeton/9780691175782.001.0001

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The Hardest Problems in NP

The Hardest Problems in NP

(p.51) Chapter 4 The Hardest Problems in NP
The Golden Ticket

Lance Fortnow

Princeton University Press

This chapter looks at some of the hardest problems in NP. Most of the NP problems that people considered in the mid-1970s either turned out to be NP-complete or people found efficient algorithms putting them in P. However, some NP problems refused to be so nicely and quickly characterized. Some would be settled years later, and others are still not known. These NP problems include the graph isomorphism, one of the few problems whose difficulty seems somewhat harder than P but not as hard as NP-complete problems like Hamiltonian paths and max-cut. Other NP problems include prime numbers and factoring, and linear programming. The linear programming problem has good algorithms in theory and practice—they just happen to be two very different algorithms.

Keywords:   NP problems, NP-complete, efficient algorithms, graph isomorphism, Hamiltonian paths, max-cut, prime numbers, factoring, linear programming

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