Approximating the weighted maximin dispersion problem over an $$\ell _p$$ℓp-ball: SDP relaxation is misleading

Consider the problem of finding a point in a unit n-dimensional $$\ell _p$$ℓp-ball ($$p\ge 2$$p≥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 $$\frac{1-O\left( \sqrt{ \ln (m)/n}\right) }{2}$$1-Oln(m)/n2, 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.