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.
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Notes
Note that \(S_i \subseteq S_j\) is the condition for the general case; for the case where \(k_i=k_j\) for all \(i,j \in \{1,2,\ldots ,n\}\), the condition is \(S_i=S_j\).
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Acknowledgements
The authors would like to thank Samuel Barnes and Nate Devine for their helpful corrections on an earlier version of this work.
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Christman, A., Chung, C., Jaczko, N. et al. Robustly assigning unstable items. J Comb Optim 44, 1556–1577 (2022). https://doi.org/10.1007/s10878-019-00515-w
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DOI: https://doi.org/10.1007/s10878-019-00515-w