Skip to Main Content
Have library access? Log in through your library
The Golden Ticket

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

Copyright Date: 2013
Pages: 200
  • Cite this Item
  • Book Info
    The Golden Ticket
    Book Description:

    The P-NP problem is the most important open problem in computer science, if not all of mathematics.The Golden Ticketprovides a nontechnical introduction to P-NP, its rich history, and its algorithmic implications for everything we do with computers and beyond. In this informative and entertaining book, Lance Fortnow traces how the problem arose during the Cold War on both sides of the Iron Curtain, and gives examples of the problem from a variety of disciplines, including economics, physics, and biology. He explores problems that capture the full difficulty of the P-NP dilemma, from discovering the shortest route through all the rides at Disney World to finding large groups of friends on Facebook. But difficulty also has its advantages. Hard problems allow us to safely conduct electronic commerce and maintain privacy in our online lives.

    The Golden Ticketexplores what we truly can and cannot achieve computationally, describing the benefits and unexpected challenges of the P-NP problem.

    eISBN: 978-1-4008-4661-0
    Subjects: Technology, Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Preface
    (pp. ix-xii)
    Lance Fortnow
  4. Chapter 1 THE GOLDEN TICKET
    (pp. 1-10)

    A candy manufacturer decides to run a contest and places a handful of golden tickets inside its chocolate bars, hidden among the tens of millions of bars produced each year. The finders of these tickets will get a rare factory tour.

    How do you find those tickets? You could buy up as many of the chocolate bars as you can. You can try using a magnet, but gold is not magnetic. Or you could hire thousands of people and give each of them a small stack of chocolate to sift through. It sounds silly, but not for Veruca Salt, who...

    (pp. 11-28)

    Imagine you are asked to write an essay that captures the social changes caused by the Internet over the past twenty years. Will you write about a device in your pocket that gives you access to all public information instantaneously? Or about how social networks connect us in entirely new ways? Or will you talk about the massive changes in the music, movie, publishing, and news businesses? One essay cannot begin to do justice to the changes that have occurred over the past two decades. Now imagine writing that chapter in the 1990s before the changes actually happened.

    If it...

  6. Chapter 3 P AND NP
    (pp. 29-50)

    Nowhere can we understand P and NP better than in Frenemy, an imaginary world where every pair of people comprises either friends or enemies.

    Frenemy has about 20,000 inhabitants. Every individual seems normal, but put two in close vicinity and a strange thing happens. Either the two take an instant liking to each other, immediately becoming the best of friends, or they take one look at each other and immediately become the worst of enemies. Despite the Frenemy name, two inhabitants are never observed taking a middle ground; they are always either close friends or distant enemies.

    These relationships appear...

    (pp. 51-70)

    Tom Hull, chair of the University of Toronto Computer Science Department in 1970, wanted to hire Steve Cook. Steve Cook had just been denied tenure at the University of California at Berkeley. Cook enjoyed sailing, so Hull took him out on Lake Ontario to show that sailing near Toronto was just as good as sailing in San Francisco Bay. The ploy worked, and Steve Cook joined the University of Toronto faculty in the fall of 1970. A brilliant move, since Cook would soon become Canada’s most famous computer scientist.

    Cook studied the connections between logic and computer science. That fall...

    (pp. 71-88)

    In the last chapter we recounted Donald Knuth’s ultimately unsuccessful attempt to find a good English word to capture NP-completeness. Knuth could have turned east to the Russians to find perebor (Перебор).Perebormeans “brute force search,” the process of trying all possibilities to find the best solution. P versus NP asks whether we need perebor to solve the clique problem or whether some faster approach could work.

    But Knuth and others in America couldn’t so easily look toward Russia. An Iron Curtain separated Russia and Eastern Europe from the United States and Western Europe starting at the end of...

    (pp. 89-108)

    In chapter 2 we saw the beautiful world where P = NP, and life was good. We could optimize everything and learn anything. We had machines that could do just about anything we could imagine them to. Very beautiful, Perhaps a little scary, and almost certainly a fantasy.

    More likely we live in a dirtier world, the world where P ≠ NP, an “inelegant universe.” Even if P = NP, until we find the algorithm that solves NP problems we might as well be living in the world of P ≠ NP. What about all those NP problems that we...

  10. Chapter 7 PROVING P ≠ NP
    (pp. 109-122)

    Juris hartmanis, one of the founders of computational complexity, has a saying: “We all know that P is different from NP, we just don’t know how to prove it.”

    In the earlier chapters we explored the P versus NP problem, what it is and why it matters, the beautiful but unlikely world where P = NP, and how to deal with hard problems if P and NP are different.

    The P versus NP problem is also an amazing challenging mathematical question. Almost immediately after Cook, Karp, and Levin brought the problem and its importance to the world, computer scientists and...

  11. Chapter 8 SECRETS
    (pp. 123-142)

    We all have secrets, from passwords to emails we don’t want the world to see. If P ≠ NP, then some NP problems have secrets, solutions that we can’t find quickly. In 1976, Whitfield Diffie and Martin Hellman suggested that we could use NP to hide our own secrets. The field of cryptography, the study of secret messages, changed forever.

    People have been sending secret messages as long as there have been messages to send. Julius Caesar used a simple substitution cipher in which each letter is replaced by the one three letters later.

    “The early bird gets the worm”...

  12. Chapter 9 QUANTUM
    (pp. 143-154)

    In 1982, the Nobel Prize-winning physicist Richard Feynman noticed there was no simple way of simulating quantum physical systems using digital computers. He turned this problem into an opportunity—perhaps a computational device based on quantum mechanics could solve problems more efficiently than more traditional computers. In the decades that followed, computer scientists and physicists, often working together, showed in theory that quantum computers can solve certain problems, such as factoring numbers, much faster. Whether we can actually build large or even medium-scale working quantum computers and determine exactly what these computers can or cannot do still remain significant challenges....

  13. Chapter 10 THE FUTURE
    (pp. 155-162)

    I’ve resigned myself to bleak outlook for the P versus NP problem: I expect that P ≠ NP and that I will not see a proof in my lifetime. We won’t see the beautiful world of chapter 1, but neither can we rule it out. The P versus NP problem will remain a mystery for decades and possibly centuries to come.

    The P versus NP problem is more than a mathematical oddity. Even if we can’t directly solve it, the P versus NP question gives us a common framework to think about and tackle the great computational tasks we need...

  14. Acknowledgments
    (pp. 163-164)
  15. Chapter Notes and Sources
    (pp. 165-170)
  16. Index
    (pp. 171-176)