Abstract
In this paper, we consider the lower-bounded k-median problem (LB k-median) that extends the classical k-median problem. In the LB k-median, a set of facilities, a set of clients and an integer k are given. Every facility has its own lower bound on the minimum number of clients that must be connected to the facility if it is opened. Every facility-client pair has its connection cost. We want to open at most k facilities and connect every client to some opened facility, such that the total connection cost is minimized.
As our main contribution, we study the LB k-median and present our main bi-criteria approximation algorithm, which, for any given constant \(\alpha \in [0,1)\), outputs a solution that satisfies the lower bound constraints by a factor of \(\alpha \) and has an approximation ratio of \(\frac{1+\alpha }{1-\alpha } \rho \), where \(\rho \) is the state-of-art approximation ratio for the k-facility location problem (k-FL). Then, by extending the main algorithm to several general versions of the LB k-median, we show the versatility of our algorithm for the LB k-median. Last, through providing relationships between the constant \(\alpha \) and the approximation ratios, we demonstrate the performances of all the algorithms for the LB k-median and its generalizations.
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Acknowledgement
The first author is supported by National Natural Science Foundation of China (No. 11531014). The second author is supported by National Natural Science Foundation of China (No. 11771003). The third author is supported by Natural Science Foundation of China (No. 11971349). The fourth author is supported by National Natural Science Foundation of China (No. 11871081) and the Science and Technology Program of Beijing Education Commission (No. KM201810005006).
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Han, L., Hao, C., Wu, C., Zhang, Z. (2020). Approximation Algorithms for the Lower-Bounded k-Median and Its Generalizations. In: Kim, D., Uma, R., Cai, Z., Lee, D. (eds) Computing and Combinatorics. COCOON 2020. Lecture Notes in Computer Science(), vol 12273. Springer, Cham. https://doi.org/10.1007/978-3-030-58150-3_51
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