A comment on “computational complexity of stochastic programming problems”


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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  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.


  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 

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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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  • Stochastic programming
  • Complexity theory
  • Two-stage problems

Mathematics Subject Classification

  • 90C15