The NP-completeness column: An ongoing guide
Journal of Algorithms
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Relaxed verification for continuous problems
Journal of Complexity
Handbook of combinatorics (vol. 2)
Computing the discrepancy with applications to supersampling patterns
ACM Transactions on Graphics (TOG)
Efficient algorithms for computing the L2-discrepancy
Mathematics of Computation
Computing discrepancies of Smolyak quadrature rules
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
Application of Threshold-Accepting to the Evaluation of the Discrepancy of a Set of Points
SIAM Journal on Numerical Analysis
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Algorithmics for Hard Problems
Algorithmics for Hard Problems
A note on E. Thiémard's algorithm to compute bounds for the star discrepancy
Journal of Complexity
Bounds and constructions for the star-discrepancy via δ-covers
Journal of Complexity
Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy
Journal of Complexity
Monte Carlo and Quasi-Monte Carlo Methods 2006
Monte Carlo and Quasi-Monte Carlo Methods 2006
Algorithmic construction of low-discrepancy point sets via dependent randomized rounding
Journal of Complexity
The complexity of geometric problems in high dimension
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Hardness of discrepancy computation and ε-net verification in high dimension
Journal of Complexity
A New Randomized Algorithm to Approximate the Star Discrepancy Based on Threshold Accepting
SIAM Journal on Numerical Analysis
Constructing low star discrepancy point sets with genetic algorithms
Proceedings of the 15th annual conference on Genetic and evolutionary computation
Uniform point sets and the collision test
Journal of Computational and Applied Mathematics
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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].