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
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.
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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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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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DOI: https://doi.org/10.1007/s10107-015-0958-2