Abstract
The work considers three problems of searching for two disjoint subsets among a finite set of points in Euclidean space. In all three problems, it is required to maximize the minimal size of these subsets so that in each cluster, the total intra-cluster scatter of points relative to the cluster center does not exceed a predetermined threshold. In the first problem, the centers of the clusters are fixed points of Euclidean space and are given as input. In the second one, centers are unknown, but they belong to the initial set. In the last problem, the center of the cluster is the arithmetic mean of all its elements. Earlier works considered problems with constraints on the quadratic intra-cluster scatter.
Quadratic analogs of the first two problems were proven to be NP-hard even in the one-dimensional case. For the third analog, the complexity remains unknown. The main result of the work are proofs of NP-hardness of all considered problems even in the one-dimensional case.
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Acknowledgments
The study presented was supported by the Russian Foundation for Basic Research, project 19-01-00308, 19-07-00397, and by the Russian Academy of Science (the Program of basic research), project 0314-2019-0015.
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Khandeev, V., Neshchadim, S. (2021). Max-Min Problems of Searching for Two Disjoint Subsets. In: Olenev, N.N., Evtushenko, Y.G., Jaćimović, M., Khachay, M., Malkova, V. (eds) Optimization and Applications. OPTIMA 2021. Lecture Notes in Computer Science(), vol 13078. Springer, Cham. https://doi.org/10.1007/978-3-030-91059-4_17
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