Skip to main content

Integer Programming: Optimization and Evaluation Are Equivalent

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 5664)

Abstract

We show that if one can find the optimal value of an integer linear programming problem in polynomial time, then one can find an optimal solution in polynomial time. We also present a proper generalization to (general) integer programs and to local search problems of the well-known result that optimization and augmentation are equivalent for 0/1-integer programs. Among other things, our results imply that PLS-complete problems cannot have “near-exact” neighborhoods, unless PLS = P.

Keywords

  • Feasible Solution
  • Polynomial Time
  • Integer Program
  • Integer Programming Problem
  • Neighborhood Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-03367-4_45
  • Chapter length: 11 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-03367-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)

    MATH  Google Scholar 

  2. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Heidelberg (1999)

    CrossRef  MATH  Google Scholar 

  3. Chakravarti, N., Wagelmans, A.P.M.: Calculation of stability radii for combinatorial optimization problems. Operations Research Letters 23, 1–7 (1998)

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Crescenzi, P., Silvestri, R.: Relative complexity of evaluating the optimum cost and constructing the optimum for maximization problems. Information Processing Letters 33, 221–226 (1990)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure Nash equilibria. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, pp. 604–612 (2004)

    Google Scholar 

  6. Grötschel, M., Lovász, L.: Combinatorial optimization. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, ch. 28, vol. 2, pp. 1541–1597. Elsevier, Amsterdam (1995)

    Google Scholar 

  7. Gutin, G., Yeo, A., Zverovitch, A.: Exponential neighborhoods and domination analysis for the TSP. In: Gutin, G., Punnen, A.P. (eds.) The Traveling Salesman Problem and Its Variations, ch. 6, pp. 223–256. Kluwer, Dordrecht (2002)

    Google Scholar 

  8. Johnson, D.S.: The NP-completeness column: Finding needles in haystacks. ACM Transactions on Algorithms 3 (2007)

    Google Scholar 

  9. Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How easy is local search? Journal of Computer and System Sciences 37, 79–100 (1988)

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Krentel, M.W., Structure in locally optimal solutions, in Proceedings of the 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, NC, 1989, 216–221.

    Google Scholar 

  11. Orlin, J.B., Punnen, A.P., Schulz, A.S.: Approximate local search in combinatorial optimization. SIAM Journal on Computing 33, 1201–1214 (2004)

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs (1982)

    MATH  Google Scholar 

  13. Ramaswamy, R., Chakravarti, N.: Complexity of determining exact tolerances for min-sum and min-max combinatorial optimization problems, Working Paper WPS-247/95, Indian Institute of Management, Calcutta, India (1995)

    Google Scholar 

  14. Schäffer, A.A., Yannakakis, M.: Simple local search problems that are hard to solve. SIAM Journal on Computing 20, 56–87 (1991)

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)

    MATH  Google Scholar 

  16. Schulz, A.S.: On the relative complexity of 15 problems related to 0/1-integer programming. In: Cook, W.J., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, ch. 19, pp. 399–428. Springer, Berlin (2009)

    CrossRef  Google Scholar 

  17. Schulz, A.S., Weismantel, R., Ziegler, G.M.: 0/1-integer programming: Optimization and augmentation are equivalent. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 473–483. Springer, Heidelberg (1995)

    CrossRef  Google Scholar 

  18. van Hoesel, S., Wagelmans, A.P.M.: On the complexity of postoptimality analysis of 0/1 programs. Discrete Applied Mathematics 91, 251–263 (1999)

    MathSciNet  CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Orlin, J.B., Punnen, A.P., Schulz, A.S. (2009). Integer Programming: Optimization and Evaluation Are Equivalent. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_45

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-03367-4_45

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03366-7

  • Online ISBN: 978-3-642-03367-4

  • eBook Packages: Computer ScienceComputer Science (R0)