ar X iv : 0 90 9 . 31 27 v 1 [ cs . C G ] 1 6 Se p 20 09 On the largest empty axis - parallel box amidst n points

We give the first nontrivial upper and lower bounds on the maximum volume of an empty axisparallel box inside an axis-parallel unit hypercube in R containing n points. For a fixed d, we show that the maximum volume is of the order Θ( 1 n ). We then use the fact that the maximum volume is Ω( 1 n ) in our design of the first efficient (1 − ε)-approximation algorithm for the following problem: Given an axis-parallel d-dimensional box R in R containing n points, compute the maximum-volume empty axis-parallel d-dimensional box contained in R. The running time of our algorithm is nearly linear in n, for small d. No previous efficient exact or approximation algorithms were known for this problem for d ≥ 4. Confirming our intuition and this status quo, recently Backer and Keil [5] have proved that the problem is NP-hard in arbitrary high dimensions (i.e., when d is part of the input). Department of Computer Science, University of Wisconsin–Milwaukee, WI 53201-0784, USA. Email: [email protected]. Supported in part by NSF CAREER grant CCF-0444188. Part of the research by this author was done at the Ecole Polytechnique Fédérale de Lausanne. Department of Computer Science, Utah State University, Logan, UT 84322-4205, USA. Email: [email protected]. Supported in part by NSF grant DBI-0743670.