Abstract
Consider the problem of finding a point in a unit n-dimensional \(\ell _p\)-ball (\(p\ge 2\)) such that the minimum of the weighted Euclidean distance from given m points is maximized. We show in this paper that the recent SDP-relaxation-based approximation algorithm (Haines et al., SIAM J Optim 23(4):2264–2294, 2013) will not only provide the first theoretical approximation bound of \(\frac{1-O\left( \sqrt{ \ln (m)/n}\right) }{2}\), but also perform much better in practice, if the SDP relaxation is removed and the optimal solution of the SDP relaxation is replaced by a simple scalar matrix.
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Notes
Their algorithm is actually proposed for the weighted maximin dispersion problem with a more general constraint.
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This research was supported by National Natural Science Foundation of China under Grants 11571029 and 11471325, and by fundamental research funds for the Central Universities under Grant YWF-17-BJ-Y-52.
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Wu, Z., Xia, Y. & Wang, S. Approximating the weighted maximin dispersion problem over an \(\ell _p\)-ball: SDP relaxation is misleading. Optim Lett 12, 875–883 (2018). https://doi.org/10.1007/s11590-017-1177-y
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DOI: https://doi.org/10.1007/s11590-017-1177-y