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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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