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Bulk-Robust combinatorial optimization

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Abstract

We commence an algorithmic study of Bulk-Robustness, a new model of robustness in combinatorial optimization. Unlike most existing models, Bulk-Robust combinatorial optimization features a highly nonuniform failure model. Instead of an interdiction budget, Bulk-Robust counterparts provide an explicit list of interdiction sets, comprising the admissible set of scenarios, thus allowing to model correlations between failures of different components in the system, interdiction sets of variable cardinality and more. The resulting model is suitable for capturing failures of complex structures in the system. We provide complexity results and approximation algorithms for Bulk-Robust counterparts of the Minimum Matroid Basis problems and the Shortest Path problem. Our results rely on various techniques, and outline the rich and heterogeneous combinatorial structure of Bulk-Robust optimization.

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Notes

  1. Recall that \(\mathrm {DTIME}(f(n))\) is the class of all decision problems that can be solved by a deterministic Turing Machine with running time \(O(f(n))\) (see e.g. [21]). The statement \(\mathrm {NP} \subseteq \mathrm {DTIME}(n^{\log \log n})\) would imply that NP–hard problems admit algorithms with running time \(O(n^{\log \log n})\). However, it is widely believed that \(\mathrm {NP}\)-hard problems do not admit quasi-polynomial time algorithms, which are algorithms with running time \(O(2^{\log ^c n})\), where \(c\) is constant.

  2. Furthermore, if one assumes that the monotone submodular function is only accessible through a value oracle, then Nemhauser and Wolsey [30] showed that any algorithm with an approximation factor of \(1-\frac{1}{e}+\epsilon \), for any fixed \(\epsilon >0\), needs an exponential number of queries to the value oracle.

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Acknowledgments

We are grateful to two referees, whose comments and suggestions considerably helped to improve the presentation of the results.

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Authors and Affiliations

  1. Institute for Operations Research, ETH Zürich, Rämistrasse 101, 8092 , Zurich, Switzerland

    David Adjiashvili

  2. Department of Mathematics, Berlin Institute of Technology, Strasse des 17. Juni 136, 10623 , Berlin, Germany

    Sebastian Stiller

  3. Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 , Zurich, Switzerland

    Rico Zenklusen

  4. Department of Applied Mathematics and Statistics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD, 21218, USA

    Rico Zenklusen

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  1. David Adjiashvili
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Correspondence to David Adjiashvili.

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Adjiashvili, D., Stiller, S. & Zenklusen, R. Bulk-Robust combinatorial optimization. Math. Program. 149, 361–390 (2015). https://doi.org/10.1007/s10107-014-0760-6

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