Let 𝒜 be a set of m points in ℝn. We show that the problem of (1 + ϵ)n-rounding of 𝒜, i.e., the problem of computing an ellipsoid E ⊆ ℝn such that [(1 + ϵ)n]−1E ⊆ conv. hull(𝒜) ⊆ E, can be solved in O(mn2(ϵ−1 + ln n + ln ln m)) arithmetic operations and comparisons. This result implies that the problem of approximating the minimum volume ellipsoid circumscribed about 𝒜 can be solved in O(m3.5 ln(mϵ−1)) operations to a relative accuracy of ϵ in the volume. The latter bound also applies to the (1 + ϵ)n-rounding problem. Our bounds hold for the real number model of computation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.