Discrepancy measures how uniformly distributed a point set is with respect to a given set of ranges. There are two notions of discrepancy, namely continuous discrepancy and combinatorial discrepancy. Depending on the ranges, several possible variants arise, for example star discrepancy, box discrepancy, and discrepancy of half-spaces. In this paper, we investigate the hardness of these problems with respect to the dimension d of the underlying space. All these problems are solvable in time n^O(d) , but such a time dependency quickly becomes intractable for high-dimensional data. Thus it is interesting to ask whether the dependency on d can be moderated. We answer this question negatively by proving that the canonical decision problems are W[1]-hard with respect to the dimension. This is done via a parameterized reduction from the Clique problem. As the parameter stays linear in the input parameter, the results moreover imply that these problems require n^ (d) time, unless 3-Sat can be solved in 2^o(n) time. Further, we derive that testing whether a given set is an -net with respect to half-spaces takes n^ (d) time under the same assumption. As intermediate results, we discover the W[1]-hardness of other well known problems, such as determining the largest empty star inside the unit cube. For this, we show that it is even hard to approximate within a factor of 2^n .
The problem has been studied extensively in the planar case, see for example @cite_24 and references therein. When the dimension is part of the input, the problem has only recently been shown to be @math -hard by @cite_11 and the fastest exact algorithm runs in time @math @cite_11 . Also recently, @cite_7 gave an @math -time @math -approximation algorithm for this problem. Note that, since @math for some @math , this counts as fpt time in parameters @math and @math , in contrast to our results for . The NP-hardness of the problem was shown by @cite_4 .
@cite_24 @cite_4 @cite_7 @cite_11
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We provide two algorithms for solving the following problem: Given a rectangle containing n points, compute the largest-area and the largest-perimeter subrectangles with sides parallel to the given rectangle that lie within this rectangle and that do not contain any points in their interior. For finding the largest-area empty rectangle, the first algorithm takes O ( n log 3 n ) time and O ( n ) memory space and it simplifies the algorithm given by Chazelle, Drysdale and Lee which takes O ( n log 3 n ) time but O ( n log n ) storage. The second algorithm for computing the largest-area empty rectangle is more complicated but it only takes O ( n log 2 n ) time and O ( n ) memory space. The two algorithms for computing the largest-area rectangle can be modified to compute the largest-perimeter rectangle in O ( n log 2 n ) and O ( n log n ) time, respectively. Since O( n log n ) is a lower bound on time for computing the largest-perimeter empty rectangle, the second algorithm for computing such a rectangle is optimal within a multiplicative constant. 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]. We give the first efficient (1−e)-approximation algorithm for the following problem: Given an axis-parallel d-dimensional box R in ℝ d containing n points, compute a maximum-volume empty axis-parallel d-dimensional box contained in R. The minimum of this quantity over all such point sets is of the order ( ( 1 n ) ). Our algorithm finds an empty axis-aligned box whose volume is at least (1−e) of the maximum in O((8ede −2) d ⋅nlog d n) time. No previous efficient exact or approximation algorithms were known for this problem for d≥4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions (i.e., when d is part of the input), the existence of an efficient exact algorithm is unlikely. The maximum empty rectangle problem is as follows: Given a set of red points in ℝd and an axis-aligned hyperrectangle B, find an axis-aligned hyperrectangle R of greatest volume that is contained in B and contains no red points. In addition to this problem, we also consider three natural variants: where we find a hypercube instead of a hyperrectangle, where we try to contain as many blue points as possible instead of maximising volume, and where we do both. Combining the results of this paper with previous results, we now know that all four of these problems (a) are NP-complete if d is part of the input, (b) have polynomial-time sweep-plane solutions for any fixed d≥3, and (c) have near linear time solutions in two dimensions.