Finding optimal volume subintervals with k points and calculating the star discrepancy are NP-hard problems

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 d-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 n points in the d-dimensional unit cube, find a subinterval of minimum or maximum volume that contains k of the n points. Our results for the complexity of the subinterval problems settle a conjecture of E. Thiemard [E. Thiemard, Optimal volume subintervals with k points and star discrepancy via integer programming, Math. Meth. Oper. Res. 54 (2001) 21-45].

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