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How to Solve NP-Complete Problems

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Book cover Limits of Computation

Part of the book series: Undergraduate Topics in Computer Science ((UTICS))

Abstract

There are hundreds of important problems that have been shown to be NP-complete. If we believe that P \(\ne \) NP, then no efficient algorithms exist to solve them. In this chapter we discuss how this affects the working programmer. As many of these NP-complete problems are relevant to business and industry, one needs coping strategies, which are the topic of this chapter. It is discussed what exact and approximation algorithms, can achieve and how parallelism and randomisation might help. Some new complexity classes will be defined along the way. The effectivity of the presented strategies are evaluated for the Travelling Salesman problem. Finally, it is discussed how the fact that there are no efficient algorithms known to factorise a number can be viewed as good news.

If there are no known feasible algorithms for NP-complete problems how do we deal with them in practice?

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Notes

  1. 1.

    In the literature one also finds the synonym tractable.

  2. 2.

    This terminology has been use in Chap. 5 of the popular science book “Computers LTD” [20].

  3. 3.

    The class of function problems corresponding to NP is usually called NPO for NP-optimisation problem.

  4. 4.

    The term “approximation algorithm” has been introduced in [24] (according to [25]).

  5. 5.

    Vazirani opens with this quote [50, Preface]:

    Although this may seem a paradox, all exact science is dominated by the idea of approximation.

    Bertrand Russell (1872–1970)

  6. 6.

    Independent Set is explained in Exercise 1.

  7. 7.

    More details, also about the history of this theorem can be found e.g. in [4].

  8. 8.

    One can drop r from the input by running the algorithm probabilistically over a uniformly distributed random bit vector r.

  9. 9.

    Sanjeev Arora, Uriel Feige, Shafi Goldwasser, Carsten Lund, László Lovász, Rajeev Motwani, Shmuel Safra, Madhu Sudan, and Mario Szegedy won the 2001 Gödel Prize for the PCP theorem and its applications to (hardness of) approximation.

  10. 10.

    See Exercise 12 in Chap. 20.

  11. 11.

    Better a program executed in parallel.

  12. 12.

    The Coppersmith-Winograd algorithm [12] does it in \(\mathop {}\mathopen {}\mathscr {O}\mathopen {}\left( n^{2.376}\right) \) but is only better for extremely large matrices and thus not used in practice, nor are recent slight improvements by Stothers or Williams.

  13. 13.

    We don’t present details in this introductory book but this will be covered in many textbooks listed in the appendix.

  14. 14.

    Chip manufacturers have to go down this route as Moore’s law is reaching its limits, as further miniaturisation of processors leads to transistors of almost atomic size and thus to unwanted and interfering quantum effects. We’ll say more about that in Chap. 23.

  15. 15.

    Michael Oser Rabin (born September 1, 1931), is an Israeli computer scientist who won the Turing-Award in 1976.

  16. 16.

    This paper has an amazing citation count of over 34,000—more than any other publication cited in this book.

  17. 17.

    The decision problem versions are mainly for complexity class analysis and not for practical purposes.

  18. 18.

    Actually the capital plus one city from each state of the United States had been chosen, which had only 48 states at the time since Alaska and Hawaii joined the United States only in 1959.

  19. 19.

    At the time of writing.

  20. 20.

    They actually show this result for other NP-complete problems as well, not just TSP.

  21. 21.

    Called PTAS reduction as it preserves the property that a problem has a polynomial time approximation scheme (PTAS).

  22. 22.

    In order to model points on a plane choose \(d=2\) (two-dimensional vector space) and in order to model points in space choose \(d=3\).

  23. 23.

    Sanjeev Arora (born January 1968) is an Indian American theoretical computer scientist who won the Gödel Prize twice: in 2001 for his work on probabilistically checkable proofs and again in 2010 for the discovery of the PTAS for EuclideanTSP.

  24. 24.

    Joseph S.B. Mitchell is an American mathematician working in computational geometry and distinguished SUNY professor. He won the Gödel prize for the discovery of the PTAS for EuclideanTSP in 2010.

  25. 25.

    Including research bases in Antarctica.

  26. 26.

    Length 7,515,772,212.

  27. 27.

    They won the Turing Award in 2002.

  28. 28.

    Bailey Whitfield ‘Whit’ Diffie (born June 5, 1944) is an American cryptographer who also worked for Sun Microsystems. He is one of the inventors of public-key cryptography for which he has won several prizes.

  29. 29.

    Martin Edward Hellman (born October 2, 1945) is an American cryptographer, and one of the inventors of public key cryptography for which he won numerous awards.

  30. 30.

    Ralph C. Merkle (born February 2, 1952) is a computer scientist and was a PhD student of Martin Hellman. He is one of the inventors of public key cryptography, for which he was awarded numerous prizes.

  31. 31.

    The Institute of Electrical and Electronics Engineers (IEEE) has granted its 100th Milestone Award to the three scientists and a plaque has been also revealed at Cheltenham in 2010.

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  1. Department of Informatics, School of Engineering and Informatics, University of Sussex, Brighton, UK

    Bernhard Reus

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Reus, B. (2016). How to Solve NP-Complete Problems. In: Limits of Computation. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-27889-6_21

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