Abstract
In a clustering with outliers problem, we are required to cluster all but a specified number of points, called outliers. In a fault tolerant clustering problem, the objective function incorporates the distance of a point to its f-th closest center chosen in the solution. We combine these two orthogonal generalizations, and consider Fault Tolerant Clustering with Outliers problems for various clustering objectives, such as k-center, k-median, and sum of radii. We essentially reduce the Fault Tolerant Clustering with Outliers problem, to the corresponding (non Fault Tolerant) Clustering with Outliers problem, for which constant approximations are known. This can be seen as a generalization of the framework of Kumar and Raichel [20] to handle the presence of outliers. This reduction comes at a loss in the approximation guarantee; however, we show that it is bounded by O(1) for the k-center objective, whereas it is O(f) for k-median and sum of radii objectives, where f is the degree of Fault Tolerance required in the solution. This implies O(1) and O(f) approximations for these generalizations respectively.
Supported by the National Science Foundation under grant CCF-1615845.
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Notes
- 1.
Charikar et al. [8] use the adjective “robust” for the generalization described here, and use the term “clustering with outliers” for the prize-collecting versions. Nevertheless, we adopt the aforementioned convention, which is otherwise standard in the literature.
- 2.
In fact, Ahmadian and Swamy [1] give a constant factor approximation for a generalization of k-clustering with lower bounds and outliers.
- 3.
For the k-clustering objective, the last step takes some more work, since we also have to compute a radius assignment \(r: Y' \rightarrow \mathbb {R}^+\). However, at a high level, the strategy is similar.
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Inamdar, T., Varadarajan, K. (2020). Fault Tolerant Clustering with Outliers. In: Bampis, E., Megow, N. (eds) Approximation and Online Algorithms. WAOA 2019. Lecture Notes in Computer Science(), vol 11926. Springer, Cham. https://doi.org/10.1007/978-3-030-39479-0_13
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