Abstract
We study the Robust Assignment Problem where the goal is to assign items of various types to containers without exceeding container capacity. We seek an assignment that uses the fewest number of containers and is robust, that is, if any item of type \(t_i\) becomes corrupt causing the containers with type \(t_i\) to become unstable, every other item type \(t_j \ne t_i\) is still assigned to a stable container. We begin by presenting an optimal polynomial-time algorithm that finds a robust assignment using the minimum number of containers for the case when the containers have infinite capacity. Then we consider the case where all containers have some fixed capacity and give an optimal polynomial-time algorithm for the special case where each type of item has the same size. When the sizes of the item types are nonuniform, we provide a polynomial-time 2-approximation for the problem. We also prove that the approximation ratio of our algorithm is no lower than 1.813. We conclude with an experimental evaluation of our algorithm.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Note that \(S_i \subseteq S_j\) is the general condition for nonuniform k; for uniform k the condition is \(S_i=S_j\).
References
Chekuri, C., Khanna, S.: A PTAS for the multiple knapsack problem. In: Symposium on Discrete Algorithms (SODA) (2000)
Epstein, L., Levin, A.: On bin packing with conflicts. In: Erlebach, T., Kaklamanis, C. (eds.) WAOA 2006. LNCS, vol. 4368, pp. 160–173. Springer, Heidelberg (2007). https://doi.org/10.1007/11970125_13
Fleischer, L., Goemans, M.X., Mirrokni, V.S., Sviridenko, M.: Tight approximation algorithms for maximum general assignment problems. In: Proceedings of the Symposium on Discrete Algorithms (2006)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Jansen, K.: An approximation scheme for bin packing with conflicts. J. Comb. Optim. 3(4), 363–377 (1999)
Jansen, K., Öhring, S.: Approximation algorithms for time constrained scheduling. Inf. Comput. 132(2), 85–108 (1997)
Korupolu, M., Rajaraman, R.: Robust and probabilistic failure-aware placement. In: Proceedings of the Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 213–224 (2016)
Korupolu, M., Meyerson, A., Rajaraman, R., Tagiku, B.: Robust and probabilistic failure-aware placement. Math. Program. 154(1–2), 493–514 (2015)
Mills, K., Chandrasekaran, R., Mittal, N.: Algorithms for optimal replica placement under correlated failure in hierarchical failure domains. Theor. Comput. Sci. (2017, pre-print)
Rahman, R., Barker, K., Alhajj, R.: Replica placement strategies in data grid. J. Grid Comput. 6(1), 103–123 (2008)
Shmoys, D., Tardos, E.: An approximation algorithm for the generalized assignment problem. Math. Program. 62(3), 461–474 (1993)
Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Mathematische Zeitschrift 27(1), 544–548 (1928)
Stein, C., Zhong, M.: Scheduling when you don’t know the number of machines. In: Proceedings of the Symposium on Discrete Algorithms (SODA) (2018)
Stirling, J.: Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium, London (1730)
Urgaonkar, B., Rosenberg, A., Shenoy, P.: Application placement on a cluster of servers. Int. J. Found. Comput. Sci. 18(5), 1023–1041 (2007)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Christman, A., Chung, C., Jaczko, N., Westvold, S., Yuen, D.S. (2018). Robustly Assigning Unstable Items. In: Kim, D., Uma, R., Zelikovsky, A. (eds) Combinatorial Optimization and Applications. COCOA 2018. Lecture Notes in Computer Science(), vol 11346. Springer, Cham. https://doi.org/10.1007/978-3-030-04651-4_27
Download citation
DOI: https://doi.org/10.1007/978-3-030-04651-4_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04650-7
Online ISBN: 978-3-030-04651-4
eBook Packages: Computer ScienceComputer Science (R0)