# A local search approximation algorithm for a squared metric *k*-facility location problem

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## Abstract

In this paper, we introduce a squared metric *k*-facility location problem (SM-*k*-FLP) which is a generalization of the squared metric facility location problem and *k*-facility location problem (*k*-FLP). In the SM-*k*-FLP, we are given a client set \(\mathcal {C}\) and a facility set \(\mathcal {F} \) from a metric space, a facility opening cost \(f_i \ge 0\) for each \( i \in \mathcal {F}\), and an integer *k*. The goal is to open a facility subset \(F \subseteq \mathcal {F}\) with \( |F| \le k\) and to connect each client to the nearest open facility such that the total cost (including facility opening cost and the sum of squares of distances) is minimized. Using local search and scaling techniques, we offer a constant approximation algorithm for the SM-*k*-FLP.

## Keywords

Approximation algorithm Facility location Local search## Notes

### Acknowledgements

The research of the first author is supported by Higher Educational Science and Technology Program of Shandong Province (No. J15LN22). The second author is supported by Natural Science Foundation of China (No. 11531014). The fourth author is supported by Natural Science Foundation of China (No. 61672323) and Natural Science Foundation of Shandong Province (ZR2016AM28). The fifth author is supported by Beijing Excellent Talents Funding (No. 2014000020124G046). A preliminary version of the paper appeared in Proceedings of the 11th Annual International Conference on Combinatorial Optimization and Applications, Shanghai, China, 2017.

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