Complexity Theory



Complexity theory is the area of the theory of computation that deals with the study and classification of the amount of computational resources required to solve problems. The subject is intellectually exciting and central to the field of computer science as well as to understanding how complex systems outside of computer science behave and compute. Complexity theory is an active area of research, still having some of the deepest unsolved problems in mathematics and computer science.


  1. Agrawal, M., Kayal, N., and Saxena, N. PRIMES is in P. Annals of Mathematics 160, 2 (2004), 781–793.MATHMathSciNetCrossRefGoogle Scholar
  2. Aho, A. V., Hopcroft, J. E., and Ullman, J. D. The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.Google Scholar
  3. Aho, A. V., Lam, M., Sethi, R., and Ullman, J. D. Compilers: Principles, Techniques, and Tools, second edition, Addison-Wesley, 2007.Google Scholar
  4. Arora, S., and Safra, S. Probabilistic checking of proofs: a new characterization of NP, Journal of the ACM 45, 1 (1998), 70–122.MATHMathSciNetCrossRefGoogle Scholar
  5. Arora, S., Lund, C., Motwani, R., Sudan, M., and Szegedy, M. Proof verification and the hardness of approximation problems, Journal of the ACM 45, 3 (1998), 501–555.MATHMathSciNetCrossRefGoogle Scholar
  6. Arora, S., and Boaz, B. Computational Complexity: A Modern Approach, Cambridge University Press, 2009.Google Scholar
  7. Babai, L., Fortnow, L., and Lund, C. Nondeterministic exponential time has two-prover interactive protocols, In Proceedings of the 31 st Annual IEEE Symposium on Foundations of Computer Science (1990), pp. 16–25.Google Scholar
  8. Babai, L., Fortnow, L., Levin, L., and Szegedy, M. Checking computations in polylogarithmic time, In Proceedings of the 23 rd Annual ACM Symposium on Theory of Computing (1991), pp. 21–32.Google Scholar
  9. Cobham, A. The intrinsic computational difficulty of functions. In Proceedings of the International Congress for Logic, Methodology, and Philosophy of Science, Y. Bar-Hillel, Ed., North-Holland, 1964. pp. 24–30.Google Scholar
  10. Cook, S. A. The complexity of theorem-proving procedures. In Proceedings of the Third Annual ACM Conference on the Theory of Computing (1971), pp. 151–158.Google Scholar
  11. Dinur, I. The PCP theorem by gap amplification. Journal of the ACM 54, 3 (2007).Google Scholar
  12. Edmonds, J. Paths, trees, and flowers. Canadian Journal of Mathematics 17 (1965), 449–467.MATHMathSciNetCrossRefGoogle Scholar
  13. Feige, U., Goldwasser, S., Lovasz, L., Safra, S., and Szegedy, M. Interactive proofs and the hardness of approximating cliques, Journal of the ACM 43, 2 (1991), 268–292.MathSciNetCrossRefGoogle Scholar
  14. Garey, M. R., and Johnson, D. S. Computers and Intractability – A Guide to the Theory of NP-completeness. W. H. Freeman, 1979.Google Scholar
  15. Goldwasser, S., Micali, S., and Rackoff, C. The knowledge complexity of interactive proof systems. SIAM Journal on Computing (1989), 186–208.Google Scholar
  16. Hartmanis, J., and Stearns, R. E. On the computational complexity of algorithms. Transactions of the American Mathematical Society 117 (1965), 285–306.MATHMathSciNetCrossRefGoogle Scholar
  17. Håstad, J. Some optimal inapproximability results. Journal of the ACM 48, 4 (2001), 105–142.CrossRefGoogle Scholar
  18. Hennie, F. C., and Stearns, R. E. Two tape simulation of multitape Turing machines, Journal of the ACM 13, 4 (1966), 533–546.MATHMathSciNetCrossRefGoogle Scholar
  19. Johnson, D. S. Approximation algorithms for combinatorial problems. Journal of Computer and Systems Sciences 9 (1974), 256–278.MATHCrossRefGoogle Scholar
  20. Karp, R. M. Reducibility among combinatorial problems. In Complexity of Computer Computations (1972), R. E. Miller and J. W. Thatcher, Eds., Plenum Press, pp. 85–103.Google Scholar
  21. Ladner, R. On the structure of polynomial time reducibility. Journal of the ACM 22, 1 (1975), 155–171.MATHMathSciNetCrossRefGoogle Scholar
  22. Levin, L. Universal search problems (in Russian). Problemy Peredachi Informatsii 9, 3 (1973), 115–116.MATHMathSciNetGoogle Scholar
  23. Papadimitriou, C. H. Computational Complexity. Addison-Wesley, 1994.Google Scholar
  24. Papadimitriou, C. H., and Yannakakis, M. Optimization, approximation, and complexity classes. Journal of Computer and Systems Sciences 43, 3 (1991), 425–440.MATHMathSciNetCrossRefGoogle Scholar
  25. Sipser, M. Introduction to the Theory of Computation, Second Edition. Thomson, 2006.MATHGoogle Scholar
  26. Trevisan, L. Inapproximability of combinatorial optimization problems. University of California, Berkeley, 2004.Google Scholar
  27. Turing, A. On computable numbers with an application to the Entscheidungsproblem. In Proc. London Mathematical Society 42, pp. 230–265, 1936.CrossRefGoogle Scholar
  28. Vazirani, V. Approximation Algorithms. Springer, 2003.Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Columbia UniversityNew YorkUSA

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