Epsilon-nets and simplex range queries
SCG '86 Proceedings of the second annual symposium on Computational geometry
Fast algorithms for computing the largest empty rectangle
SCG '87 Proceedings of the third annual symposium on Computational geometry
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Computing the discrepancy with applications to supersampling patterns
ACM Transactions on Graphics (TOG)
On the efficiency of polynomial time approximation schemes
Information Processing Letters
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
The Maximum Box Problem and its Application to Data Analysis
Computational Optimization and Applications
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Geometric clustering: fixed-parameter tractability and lower bounds with respect to the dimension
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy
Journal of Complexity
On Approximating the Depth and Related Problems
SIAM Journal on Computing
Tight lower bounds for certain parameterized NP-hard problems
Information and Computation
Algorithmic construction of low-discrepancy point sets via dependent randomized rounding
Journal of Complexity
Tight lower bounds for the size of epsilon-nets
Proceedings of the twenty-seventh annual symposium on Computational geometry
Geometric clustering: Fixed-parameter tractability and lower bounds with respect to the dimension
ACM Transactions on Algorithms (TALG)
The mono- and bichromatic empty rectangle and square problems in all dimensions
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
A Non-linear Lower Bound for Planar Epsilon-nets
Discrete & Computational Geometry
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
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Discrepancy measures how uniformly distributed a point set is with respect to a given set of ranges. Depending on the ranges, several variants arise, including star discrepancy, box discrepancy, and discrepancy of halfspaces. These problems are solvable in time n^O^(^d^), where d is the dimension of the underlying space. As such a dependency on d becomes intractable for high-dimensional data, we ask whether it can be moderated. We answer this question negatively by proving that the canonical decision problems are W[1]-hard with respect to the dimension, implying that no f(d)@?n^O^(^1^)-time algorithm is possible for any function f(d) unless FPT=W[1]. We also discover the W[1]-hardness of other well known problems, such as determining the largest empty box that contains the origin and is inside the unit cube. This is shown to be hard even to approximate within a factor of 2^n.