On the Largest Empty Axis-Parallel Box Amidst n Points

We give the first efficient (1− ε )-approximation algorithm for the following problem: Given an axis-parallel d -dimensional box R in ℝ d containing n points, compute a maximum-volume empty axis-parallel d-dimensional box contained in R . The minimum of this quantity over all such point sets is of the order . Our algorithm finds an empty axis-aligned box whose volume is at least (1− ε ) of the maximum in O ((8 edε −2 ) d ⋅ n log d n ) time. No previous efficient exact or approximation algorithms were known for this problem for d ≥4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions ( i.e. , when d is part of the input), the existence of an efficient exact algorithm is unlikely. We also present a (1− ε )-approximation algorithm that, given an axis-parallel d -dimensional cube R in ℝ d containing n points, computes a maximum-volume empty axis-parallel hypercube contained in R . The minimum of this quantity over all such point sets is also shown to be of the order . A faster (1− ε )-approximation algorithm, with a milder dependence on d in the running time, is obtained in this case.