Abstract
In this paper, we study the adaptability of minimizing a non-increasing supermodular function f with cardinality constraint k. We first propose an algorithm with \(\mathcal {O}\left( \log ^2n \cdot \log \frac{f(S_0)}{\epsilon \cdot \mathtt {OPT}}\right) \) adaptive rounds which can return a solution set S with \(|S| = |S_0|+ O\left( k \log {\frac{f(S_0)}{\epsilon \cdot \mathtt {OPT}}}\right) \) satisfying \(f(S) \le (1+\epsilon ) \mathtt {OPT}\), where \(S_0\) is the initial solution set and \(\mathtt {OPT}\) is the optimal value. The adaptivity is then improved to \(\mathcal {O}\left( \log n \cdot \log \frac{f(S_0)}{\epsilon \cdot \mathtt {OPT}}\right) \) by the second algorithm. The application of the new algorithms to the fuzzy C-means problem is also discussed.
Supported by National Natural Science Foundation of China (No. 12001335), Shandong Provincial Natural Science Foundation, China (Nos. ZR2020MA029, ZR2019PA004).
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Zhang, X., Liu, Q., Li, M., Zhou, Y. (2021). Bi-criteria Adaptive Algorithms for Minimizing Supermodular Functions with Cardinality Constraint. In: Wu, W., Du, H. (eds) Algorithmic Aspects in Information and Management. AAIM 2021. Lecture Notes in Computer Science(), vol 13153. Springer, Cham. https://doi.org/10.1007/978-3-030-93176-6_16
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DOI: https://doi.org/10.1007/978-3-030-93176-6_16
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