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

Abstract

a b s t r a c tThe 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. Thiémard [E. Thiémard, Optimal volume subintervals with k points and star discrepancy via integer programming, Math. Meth. Oper. Res. 54 (2001) 21-45].

Algorithmic construction of low-discrepancy point sets via dependent randomized rounding

“…Because of the difficulty of determining or even usefully approximating the star discrepancy of a high-dimensional point set (see, e.g., [15,17,13]) we are quite limited in our choice of dimension and number of points for our experiments. Since theory suggests that our method gains advantage over the classical methods with increasing dimension, we still wish to evaluate our point sets for higher dimensions, and so we are forced to use gradually less precise methods as d and n increase.…”

“…This approach sounds simple, but overlooks the difficulty of calculating (or approximating) high-dimensional star discrepancies. Indeed, it has been shown in [17] that the decision problem as regards whether an arbitrary point set has discrepancy smaller than ε is NP-hard, and recently this was improved to showing that under certain complexity theoretical assumptions, the running time must essentially scale as n Θ(d) [13]. Also, all deterministic algorithms known so far that approximate the L ∞ -star discrepancy of arbitrary given n-point sets have a running time exponential in d (see, e.g., [30], the literature mentioned therein, and the discussion in [15]).…”

Discrepancy-based evolutionary diversity optimization

“…Here Vol(J) denotes the volume of interval J and Y is the class of all subintervals of the form A key difficulty to overcome in the optimization for low star discrepancy values is the computational hardness of its evaluation [12]. The best known algorithm for the star discrepancy computation has a running time of order n 1+d/2 [7], which is exponential in the dimension d. As we are interested in dimension d = 2, 3, we can use this algorithm, and make use of the implementation that is available on [29].…”

The Inverse of the Star-Discrepancy Problem and the Generation of Pseudo-Random Numbers

“…All results described so far have either been existence results of point sets with small star-discrepancy, or results for point sets with small star-discrepancy which can be obtained via computer search. The Ansatz via computer search remains difficult and is limited to a rather small number of points N and dimensions s (in fact, it is known that the computation of the star-discrepancy is N P -hard as shown by Gnewuch, Srivastav, and Winzen [19], which makes it difficult to obtain good point sets via computer search). To make the random constructions useful in applications, Aistleitner and Hofer [3] show that with probability δ one can expect point sets with discrepancy of order c(δ) s/N .…”