Article ID Journal Published Year Pages File Type
4609002 Journal of Complexity 2009 13 Pages PDF
Abstract

The well-known star discrepancy is a common measure for the uniformity of point distributions. It is used, e.g., in multivariate integration, pseudo random number generation, experimental design, statistics, or computer graphics.We study here the complexity of calculating the star discrepancy of point sets in the dd-dimensional unit cube and show that this is an NP-hard problem.To establish this complexity result, we first prove NP-hardness of the following related problems in computational geometry: Given nn points in the dd-dimensional unit cube, find a subinterval of minimum or maximum volume that contains kk of the nn points.Our results for the complexity of the subinterval problems settle a conjecture of E. Thiémard [E. Thiémard, Optimal volume subintervals with kk points and star discrepancy via integer programming, Math. Meth. Oper. Res. 54 (2001) 21–45].

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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