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The Hardest Problems in NP

Lance Fortnow

in The Golden Ticket: P, NP, and the Search for the Impossible

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175782
eISBN:
9781400846610
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175782.003.0004
Subject:
Computer Science, Programming Languages

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 ... More


Proving P ≠ NP

Lance Fortnow

in The Golden Ticket: P, NP, and the Search for the Impossible

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175782
eISBN:
9781400846610
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175782.003.0007
Subject:
Computer Science, Programming Languages

This chapter focuses on a few of the ideas that people have tried to solve the P versus NP problem. These have not panned out to anything close to a solution to the problem. To prove P ≠ NP one needs ... More


The Golden Ticket: P, NP, and the Search for the Impossible

Lance Fortnow

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175782
eISBN:
9781400846610
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175782.001.0001
Subject:
Computer Science, Programming Languages

The P versus NP problem is the most important open problem in computer science, if not all of mathematics. Simply stated, it asks whether every problem whose solution can be quickly checked by ... More


Dealing with Hardness

Lance Fortnow

in The Golden Ticket: P, NP, and the Search for the Impossible

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175782
eISBN:
9781400846610
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175782.003.0006
Subject:
Computer Science, Programming Languages

This chapter demonstrates several approaches for dealing with hard problems. These approaches include brute force, heuristics, and approximation. Typically, no single technique will suffice to handle ... More


The Golden Ticket

Lance Fortnow

in The Golden Ticket: P, NP, and the Search for the Impossible

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175782
eISBN:
9781400846610
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175782.003.0001
Subject:
Computer Science, Programming Languages

This introductory chapter provides an overview of the P versus NP problem. The P versus NP problem asks, among other things, whether one can quickly find the shortest route for a traveling salesman. ... More


The Future

Lance Fortnow

in The Golden Ticket: P, NP, and the Search for the Impossible

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175782
eISBN:
9781400846610
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175782.003.0010
Subject:
Computer Science, Programming Languages

This chapter explores some of today's great challenges of computing. These challenges include parallel computation, dealing with big data, and the networking of everything. The chapter then argues ... More


The Beautiful World

Lance Fortnow

in The Golden Ticket: P, NP, and the Search for the Impossible

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175782
eISBN:
9781400846610
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175782.003.0002
Subject:
Computer Science, Programming Languages

This chapter examines an efficient algorithm that solves NP problems, the Urbana algorithm. With the Urbana algorithm one can solve all the NP problems quickly, finding the simplest program that ... More


The algorithm: Mathematics and computers

Marcel Danesi

in Pythagoras' Legacy: Mathematics in Ten Great Ideas

Published in print:
2020
Published Online:
January 2020
ISBN:
9780198852247
eISBN:
9780191886959
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198852247.003.0010
Subject:
Mathematics, History of Mathematics, Educational Mathematics

Mathematics has formed a partnership with computer science, since at least the 1940s. Devising algorithms and turning them into computer programs is akin to deconstructing a problem, a proof, or a ... More


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