A comment on “computational complexity of stochastic programming problems”

Abstract

Although stochastic programming problems were always believed to be computationally challenging, this perception has only recently received a theoretical justification by the seminal work of Dyer and Stougie (Math Program A 106(3):423–432, 2006). Amongst others, that paper argues that linear two-stage stochastic programs with fixed recourse are #P-hard even if the random problem data is governed by independent uniform distributions. We show that Dyer and Stougie’s proof is not correct, and we offer a correction which establishes the stronger result that even the approximate solution of such problems is #P-hard for a sufficiently high accuracy. We also provide new results which indicate that linear two-stage stochastic programs with random recourse seem even more challenging to solve.

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Notes

  1. 1.

    The complexity class #P contains the counting problems associated with the decision problems in the complexity class NP (e.g., counting the number of Hamiltonian cycles in a graph), see [6, 9]. Thus, a counting problem is in #P if the items to be counted (e.g., the Hamiltonian cycles) can be validated as such in polynomial time. By definition, a #P problem is at least as difficult as the corresponding NP problem. It is therefore commonly believed that #P-hard problems, which are the hardest problems in #P, do not admit polynomial-time solution methods.

References

  1. 1.

    Balakrishnan, N., Nevzorov, V.B.: A Primer on Statistical Distributions. Wiley, Hoboken, NY (2004)

    Google Scholar 

  2. 2.

    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming Springer. Series in Operations Research. Springer, Berlin (1997)

    Google Scholar 

  3. 3.

    Brightwell, G., Winkler, P.: Counting linear extensions. Order 8(3), 225–242 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Dyer, M., Stougie, L.: Computational complexity of stochastic programming problems. Math. Program. A 106(3), 423–432 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Dyer, M.E., Frieze, A.M.: On the complexity of computing the volume of a polyhedron. SIAM J. Comput. 17(5), 967–974 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)

    Google Scholar 

  7. 7.

    Gautschi, W.: On inverses of Vandermonde and confluent Vandermonde matrices. Numer. Math. 4(1), 117–123 (1962)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Leövey, H., Römisch, W.: Quasi-Monte Carlo methods for linear two-stage stochastic programming problems. Math. Program. 151(1), 315–345 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Boston (1994)

    Google Scholar 

  10. 10.

    Prékopa, A.: Stochastic Programming. Kluwer Academic Publishers, Berlin (1995)

    Google Scholar 

  11. 11.

    Ruszczyński, A., Shapiro, A. (eds.): Stochastic Programming, Volume 10 of Handbooks in Operations Research and Management Science. Elsevier, Amsterdam (2003)

    Google Scholar 

  12. 12.

    Shapiro, A., Nemirovski, A.: On complexity of stochastic programming problems. In: Jeyakumar, V., Rubinov, A.M. (eds.) Continuous Optimization: Current Trends and Applications, pp. 111–144. Springer, Berlin (2005)

    Google Scholar 

  13. 13.

    Sloan, I.H., Woźniakowski, H.: When are Quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex. 14(1), 1–33 (1998)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the anonymous referees for their thoughtful comments which substantially improved the paper. This research was supported by the Swiss National Science Foundation Grant BSCGI0_157733 and the EPSRC Grants EP/M028240/1 and EP/M027856/1.

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Correspondence to Grani A. Hanasusanto.

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Hanasusanto, G.A., Kuhn, D. & Wiesemann, W. A comment on “computational complexity of stochastic programming problems”. Math. Program. 159, 557–569 (2016). https://doi.org/10.1007/s10107-015-0958-2

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Keywords

  • Stochastic programming
  • Complexity theory
  • Two-stage problems

Mathematics Subject Classification

  • 90C15