Abstract
This chapter introduces combinatorial optimization problems and their computational complexity. We first formulate some fundamental problems already introduced in the previous chapter and then consider basic concepts of the theory of computational complexity, with special emphasis on decision problems, polynomial-time algorithms, and NP-complete problems. The chapter concludes with a discussion of solution approaches for NP-hard problems, introducing constructive heuristics, local search or improvement procedures and, finally, metaheuristics.
References
E.H.L. Aarts and J. Korst. Simulated annealing and Boltzmann machines: A stochastic approach to combinatorial optimization and neural computing. Wiley, New York, 1989.
D. Bertsimas and R. Weismantel. Optimization over integers. Dynamic Ideas, Belmont, 2005.
E.K. Burke and G. Kendall, editors. Search methodologies: Introductory tutorials in optimization and decision support techniques. Springer, New York, 2005.
E.K. Burke and G. Kendall, editors. Search methodologies: Introductory tutorials in optimization and decision support techniques. Springer, New York, 2nd edition, 2014.
A. Cobham. The intrinsic computational difficulty of functions. In Y. Bar-Hillel, editor, Proceedings of the 1964 International Congress for Logical Methodology and Philosophy of Science, pages 24–30, Amsterdam, 1964. North Holland.
S.A. Cook. The complexity of theorem-proving procedures. In M.A. Harrison, R.B. Banerji, and J.D. Ullman, editors, Proceedings of the Third Annual ACM Symposium on Theory of Computing, pages 151–158, New York, 1971. ACM.
T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein. Introduction to Algorithms. MIT Press, Cambridge, 3rd edition, 2009.
J. Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17: 449–467, 1965.
J. Edmonds. Minimum partition of a matroid in independent subsets. Journal of Research, National Bureau of Standards, 69B:67–72, 1975.
T.A. Feo and M.G.C. Resende. Greedy randomized adaptive search procedures. Journal of Global Optimization, 6:109–133, 1995.
P. Festa and M.G.C. Resende. GRASP: An annotated bibliography. In C.C. Ribeiro and P. Hansen, editors, Essays and surveys in metaheuristics, pages 325–367. Kluwer Academic Publishers, Boston, 2002.
P. Festa and M.G.C. Resende. An annotated bibliography of GRASP, Part I: Algorithms. International Transactions in Operational Research, 16:1–24, 2009a.
P. Festa and M.G.C. Resende. An annotated bibliography of GRASP, Part II: Applications. International Transactions in Operational Research, 16, 2009b. 131–172.
M.R. Garey and D.S. Johnson. Approximation algorithms for combinatorial problems: An annotated bibliography. In J.F. Traub, editor, Algorithms and complexity: New directions and recent results, pages 41–52. Academic Press, Orlando, 1976.
M.R. Garey and D.S. Johnson. Strong NP-completeness results: Motivation, examples, and implications. Journal of the ACM, 25:499–508, 1978.
M.R. Garey and D.S. Johnson. Computers and intractability. Freeman, San Francisco, 1979.
M. Gendreau and J.-Y. Potvin, editors. Handbook of metaheuristics. Springer, New York, 2nd edition, 2010.
F. Glover and G. Kochenberger, editors. Handbook of metaheuristics. Kluwer Academic Publishers, Boston, 2003.
F. Glover and M. Laguna. Tabu search. Kluwer Academic Publishers, Boston, 1997.
D.E. Goldberg. Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Reading, 1989.
M. Grötschel, L. Lovász, and A. Schrijver. Polynomial algorithms for perfect graphs. Annals of Discrete Mathematics, 21:325–356, 1984.
H.H. Hoos and T. Stützle. Stochastic local search: Foundations and applications. Elsevier, New York, 2005.
D.S. Johnson. Near-optimal bin-packing algorithms. PhD thesis, Massachusetts Institute of Technology, Cambridge, 1973.
D.S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 9:256–278, 1974.
R.M. Karp. Reducibility among combinatorial problems. In R.E. Miller and J.W. Thatcher, editors, Complexity of computer computations. Plenum Press, New York, 1972.
R.M. Karp. On the computational complexity of combinatorial problems. Networks, 5:45–68, 1975.
M.R. Krom. The decision problem for a class of first-order formulas in which all disjunctions are binary. Zeitschrift fr Mathematische Logik und Grundlagen der Mathematik, 13:15–20, 1967.
K. Kuratowski. Sur le problème des courbes gauches en topologie. Fundamenta Mathematicae, 15:271–283, 1930.
S. Martello and P. Toth. Knapsack problems: Algorithms and computer implementations. John Wiley & Sons, New York, 1990.
Z. Michalewicz. Genetic algorithms + Data structures = Evolution programs. Springer, Berlin, 1996.
W. Michelis, E.H.L. Aarts, and J. Korst. Theoretical aspects of local search. Springer, Berlin, 2007.
G.L. Nemhauser and L.A. Wolsey. Integer and combinatorial optimization. Wiley, New York, 1988.
N.J. Nilsson. Problem-solving methods in artificial intelligence. McGraw-Hill, New York, 1971.
N.J. Nilsson. Principles of artificial intelligence. Springer, Berlin, 1982.
C.H. Papadimitriou. Computational complexity. Addison-Wesley, Reading, 1994.
C.H. Papadimitriou and K. Steiglitz. Combinatorial optimization: Algorithms and complexity. Prentice Hall, Englewood Cliffs, 1982.
J. Pearl. Heuristics: Intelligent search strategies for computer problem solving. Addison-Wesley, Reading, 1985.
L.S. Pitsoulis and M.G.C. Resende. Greedy randomized adaptive search procedures. In P.M. Pardalos and M.G.C. Resende, editors, Handbook of applied optimization, pages 168–183. Oxford University Press, New York, 2002.
C.R. Reeves. Modern heuristic techniques for combinatorial problems. Blackwell, London, 1993.
M.G.C. Resende and C.C. Ribeiro. Greedy randomized adaptive search procedures. In F. Glover and G. Kochenberger, editors, Handbook of metaheuristics, pages 219–249. Kluwer Academic Publishers, Boston, 2003b.
M.G.C. Resende and C.C. Ribeiro. GRASP with path-relinking: Recent advances and applications. In T. Ibaraki, K. Nonobe, and M. Yagiura, editors, Metaheuristics: Progress as real problem solvers, pages 29–63. Springer, New York, 2005a.
M.G.C. Resende and C.C. Ribeiro. Parallel greedy randomized adaptive search procedures. In E. Alba, editor, Parallel metaheuristics: A new class of algorithms, pages 315–346. Wiley-Interscience, Hoboken, 2005b.
M.G.C. Resende and C.C. Ribeiro. Greedy randomized adaptive search procedures: Advances and applications. In M. Gendreau and J.-Y. Potvin, editors, Handbook of metaheuristics, pages 293–319. Springer, New York, 2nd edition, 2010.
C.C. Ribeiro. GRASP: Une métaheuristique gloutonne et probabiliste. In J. Teghem and M. Pirlot, editors, Optimisation approchée en recherche opérationnelle, pages 153–176. Hermès, Paris, 2002.
A. Schrijver. Theory of linear and integer programming. Wiley, New York, 1986.
P.J.M. van Laarhoven and E. Aarts. Simulated annealing: Theory and applications. Kluwer Academic Publishers, Boston, 1987.
V.V. Vazirani. Approximation algorithms. Springer, Berlin, 2001.
D.P. Williamson and D.B. Shmoys. The design of approximation algorithms. Cambridge University Press, New York, 2011.
L.A. Wolsey. Integer programming. Wiley, New York, 1998.
M. Yannakakis. Computational complexity. In E.H.L. Aarts and J.K. Lenstra, editors, Local search in combinatorial optimization, chapter 2, pages 19–55. Wiley, Chichester, 2007.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media New York
About this chapter
Cite this chapter
Resende, M.G.C., Ribeiro, C.C. (2016). A short tour of combinatorial optimization and computational complexity. In: Optimization by GRASP. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6530-4_2
Download citation
DOI: https://doi.org/10.1007/978-1-4939-6530-4_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-6528-1
Online ISBN: 978-1-4939-6530-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)