Abstract
Stochastic combinatorial optimization problems are usually defined as planning problems, which involve purchasing and allocating resources in order to meet uncertain needs. For example, network designers need to make their best guess about the future needs of the network and purchase capabilities accordingly. Facing uncertain in the future, we either “wait and see” changes, or postpone decisions about resource allocation until the requirements or constraints become realized. Specifically, in the field of stochastic combinatorial optimization, some inputs of the problems are uncertain, but follow known probability distributions. Our goal is to find a strategy that minimizes the expected cost. In this paper, we consider the two-stage finite-scenario stochastic set cover problem and the single sink rent-or-buy problem by presenting primal-dual based approximation algorithms for these two problems with approximation ratio \(2\eta \) and 4.39, respectively, where \(\eta \) is the maximum frequency of the element of the ground set in the set cover problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arora A, Sudan M (2003) Improved low degree testing and applications. Combinatorica 23(3):365–426
Balkanski E, Singer Y (2020) A lower bound for parallel submodular minimization. In: Proceedings of the 52nd annual ACM SIGACT symposium on theory of computing (STOC), pp 130–139
Bellare M, Goldwass S, Lund C, Russell A (1994) Efficient probabilistically checkable proofs and applications to approximations. In: Proceedings of the 26th annual ACM symposium on theory of computing (STOC), pp 23–25
Byrka J, Grandon F (2010) An improved LP-based approximation for Steiner tree. In: Proceedings of the 42nd ACM symposium on theory of computing (STOC), pp 5–8
Fleischer L, Iwata S (2003) A push-relabel framework for submodular function minimization and applications to parametric optimization. Discrete Appl Math 131(2):311–322
Gandhi R, Khuller S, Srinivasan A (2004) Approximation algorithms for partial covering problems. J Algorithms 53(1):55–84
Gupta A, Kleinberg J, Kumar A, Rastogi R, Yener B (2001) Provisioning a virtual private network: a network design problem for multicommodity flow. In: Proceedings of the 33rd annual ACM symposium on theory of computing (STOC), pp 389–398
Hochbaum D-S (1982) Approximation algorithm for set covering and vertex cover problems. SIAM J Comput 11(3):555–556
Johnson D-S (1974) Approximation algorithms for combinatorial problems. J Comput Syst Sci 9(3):256–278
Karger D-R, Minkoff M (2000) Building Steiner trees with incomplete global knowledge. In: Proceedings of the 41st annual symposium on foundations of computer science (FOCS), pp 613–623
Karp R-M (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW, Bohlinger JD (eds) Complexity of computer computations. The IBM Research Symposia Series. Springer, Boston, pp 85–103
Khuller S, Zhu A (2002) The general Steiner tree-star problem. Inf Process Lett 84(4):215–220
Kearns M-J (1990) The computational complexity of machine learning. The MIT Press, Cambridge
Kim T-U, Lowe T-J, Tamir A, Ward J-E (1996) On the location of a tree-shaped facility. Networks 28(3):167–175
Könemann J, Parekh O, Segev D (2011) A unified approach to approximating partial covering problems. Algorithmica 59(4):489–509
Labbé M, Laporte G, Martin I-R, González J-J-S (2001) The median cycle problem. Technical Report 2001/12, Department of Operations Research and Multicriteria Decision Aid at Université Libre de Bruxelles
Lee Y, Chiu S-Y, Ryan J (1996) A branch and cut algorithm for a Steiner tree-star problem. Inf J Comput 8(3):194–201
Li J, Liu Y (2016) Approximation algorithms for stochastic combinatorial optimization problems. J Oper Res Soc China 4(1):1–47
Parthasarathy S (2018) Adaptive greedy algorithms for stochastic set cover problems. arXiv:1803.07639
Ravic R, Sinhac A (2006) Hedging uncertainty: approximation algorithms for stochastic optimization problems. Math Programm 108(1):97–114
Raz R, Safra S (1997) A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th annual ACM symposium on the theory of computing (STOC), pp 4–6
Slavik P (1997) Improved performance of greedy algorithm for partial over. Inf Process Lett 64(5):251–254
Swamy C, Kumar A (2004) Primal-dual algorithms for connected facility location problems. Algorithmica 40(4):245–269
Tang S (2021) Beyond pointwise submodularity: non-monotone adaptive submodular maximization in linear time. Theor Comput Sci 850:249–261
Acknowledgements
This research is supported or partially supported by the National Natural Science Foundation of China (Grant Nos. 11871280, 11871081, 11771386 and 11728104), the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant 06446, Qinglan Project and Zhejiang Provincial Natural Science Foundation (No. LY20A010013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A preliminary version appeared in the Proceedings of the 14th International Conference on Algorithmic Aspects in Information and Management (AAIM), August 10–12, 2020, Jinhua, China.
Rights and permissions
About this article
Cite this article
Sun, J., Sheng, H., Sun, Y. et al. Approximation algorithms for stochastic set cover and single sink rent-or-buy with submodular penalty. J Comb Optim 44, 2626–2641 (2022). https://doi.org/10.1007/s10878-021-00753-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-021-00753-x