Axis-parallel Boxes Research Articles

Improved bounds for the bracketing number of orthants or revisiting an algorithm of Thiémard to compute bounds for the star discrepancy

We improve the best known upper bound for the bracketing number of d-dimensional axis-parallel boxes anchored in 0 (or, put differently, of lower left orthants intersected with the d-dimensional unit cube [0,1]d). More precisely, we provide a better estimate for the cardinality of an algorithmic bracketing cover construction due to Eric Thiémard, which forms the core of his algorithm to approximate the star discrepancy of arbitrary point sets from Thiémard (2001) [22]. Moreover, the new upper bound for the bracketing number of anchored axis-parallel boxes yields an improved upper estimate for the bracketing number of arbitrary axis-parallel boxes in [0,1]d. In our upper bounds all constants are fully explicit.

Hypervolume-Guided Decomposition for Parallel Expensive Multiobjective Optimization

The hypervolume metric is widely used to guide the search in multiobjective optimization. However, in parallel expensive multiobjective optimization, the hypervolume-based multi-point expected improvement (EI) suffers from high computational overhead and scales poorly with the batch size. To address this issue, we integrate hypervolume-based EI with the MOEA/D framework and propose a novel EI, named the expected direction-based hypervolume improvement (DirHV-EI). The DirHV-EI only measures the hypervolume improvement within each axis-parallel box induced by the modified Tchebycheff scalarization. Thus, it has a simple analytical expression that can be easily computed. Theoretical analysis indicates that the maximization of our proposed improvement function can help to maximize both the weighted hypervolume and the Tchebycheff improvement metrics. Using DirHV-EI, we design a decomposition-based Bayesian optimization algorithm, called DirHV-EGO, for solving expensive multiobjective optimization problems. At each iteration, the MOEA/D is used to maximize the DirHV-EI values with respect to a number of direction vectors in a collaborative manner, and a number of candidate solutions can be obtained. Then, a submodularity-based greedy selection strategy is used to select multiple query points from the candidates. Experimental results on both benchmark instances and real-world problems show that our proposed algorithm is an efficient and effective method for parallel expensive multiobjective optimization.

Lower bounds for piercing and coloring boxes

Given a family B of axis-parallel boxes in Rd, let τ denote its piercing number, and ν its independence number. It is an old question whether τ/ν can be arbitrarily large for given d≥2. Here, for every ν, we construct a family of axis-parallel boxes achievingτ≥Ωd(ν)⋅(log⁡νlog⁡log⁡ν)d−2. This not only answers the previous question for every d≥3 positively, but also matches the best known upper bound up to double-logarithmic factors.Our main construction has further implications about the Ramsey and coloring properties of configurations of boxes as well. We show the existence of a family of n boxes in Rd, whose intersection graph has clique and independence number Od(n1/2)⋅(log⁡nlog⁡log⁡n)−(d−2)/2. This is the first improvement over the trivial upper bound Od(n1/2), and matches the best known lower bound up to double-logarithmic factors. Finally, for every ω satisfying log⁡nlog⁡log⁡n≪ω≪n1−ε, we construct an intersection graph of n boxes with clique number at most ω, and chromatic number Ωd,ε(ω)⋅(log⁡nlog⁡log⁡n)d−2. This matches the best known upper bound up to a factor of Od((log⁡ω)(log⁡log⁡n)d−2).

On the boxicity of Kneser graphs and complements of line graphs

The boxicity of a graph G=(V,E), denoted by box(G), is the minimum dimension d such that G is the intersection graph of a family (Bv)v∈V of d-dimensional axis-parallel boxes in Rd.Let k and n be two positive integers such that n≥2k+1. The Kneser graphK(n,k) is the graph with vertex set given by all subsets of {1,2,…,n} of size k where two vertices are adjacent if their corresponding k-sets are disjoint. In this note, we derive a general upper bound for box(K(n,k)), and a lower bound in the case n≥2k3−2k2+1, which matches the upper bound up to an additive factor of Θ(k2). Our second contribution is to provide upper and lower bounds for the boxicity of the complement of the line graph of any graph G, and as a corollary we derive that box(K(n,2))∈{n−3,n−2} for every n≥5.

Burling graphs revisited, part I: New characterizations

The Burling sequence is a sequence of triangle-free graphs of increasing chromatic number. Each of them is isomorphic to the intersection graph of a set of axis-parallel boxes in R3. These graphs were also proved to have other geometrical representations: intersection graphs of line segments in the plane, and intersection graphs of frames, where a frame is the boundary of an axis-aligned rectangle in the plane.We call Burling graph every graph that is an induced subgraph of some graph in the Burling sequence. We give five new equivalent ways to define Burling graphs. Three of them are geometrical, one is of a more graph-theoretical flavor, and one, that we call abstract Burling graphs, is more axiomatic.

The area of empty axis-parallel boxes amidst 2-dimensional lattice points

The dispersion of a point set in the unit square is the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the lattice generators. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subrings of the ring of integers of quadratic fields. We have tight bounds for the dispersion of a lattice based on the largest continued fraction coefficient of the generators, accurate to within one half. We provide an equivalent formulation of Zaremba's conjecture. Using this framework we are able to give very short proofs of previous results.

On generation and enumeration of orthogonal Chebyshev-Frolov lattices

We discuss orthogonal Chebyshev-Frolov lattices, their generating matrices and their use in Frolov cubature formula. We give a detailed account on coordinate-permuted systems that lead to fast computation and enumeration of such lattices. In particular, we explain the recurrences identified in (K. Suzuki and T. Yoshiki, Hiroshima Math. J., 49(1):139–159, 2019) via a plain constructive approach exhibiting a new hierarchical basis of polynomials. Dual Chebyshev-Frolov lattices and their generating matrices are also studied. Lattices enumeration in axis-parallel boxes is discussed.

Minimal dispersion of large volume boxes in the cube

In this note we present a construction which improves the best known bound on the minimal dispersion of large volume boxes in the unit cube. Let d>1. The dispersion of T⊂[0,1]d is defined as the supremum of the volume taken over all axis parallel boxes in the cube which do not intersect T. The minimal dispersion of n points in the cube is defined as the infimum of the dispersion taken over all T such that |T|=n. Define the “large volume” regime as the class of all volumes 14<r≤12. The inverse of the minimal dispersion is denoted as N(r,d). When the volume is large, the best known upper bound on N(r,d) is of the order (r−14)−1. The construction presented in this note yields an upper bound given byN(r,d)≤⌊πr−14⌋−3. Some of our intermediate estimates are sharp given the condition that d≥Cr, where Cr is a positive constant which depends only on the volume r.

New bounds on the minimal dispersion

We provide a new construction for a set of boxes approximating axis-parallel boxes of fixed volume in [0,1]d. This improves upper bounds for the minimal dispersion of a point set in the unit cube and its inverse in both the periodic and non-periodic settings in certain regimes. In the case of random choice of points our bounds are sharp up to double logarithmic factor. We also apply our construction to k-dispersion.

Geometric systems of unbiased representatives

Let P be a finite point set in Rd, B be a bicoloring of P and O be a family of geometric objects (that is, intervals, boxes, balls, etc). An object from O is called balanced with respect to B if it contains the same number of points from each color of B. For a collection B of bicolorings of P, a geometric system of unbiased representatives (G-SUR) is a subset O′⊆O such that for any bicoloring B of B there is an object in O′ that is balanced with respect to B.We pose and study problems on finding G-SURs. We obtain general bounds on the size of G-SURs consisting of intervals, size-restricted intervals, axis-parallel boxes and Euclidean balls. We show that the G-SUR problem is NP-Hard even in the simple case of points on a line and interval ranges. Furthermore, we study a related problem on determining the size of the largest and smallest balanced intervals for points on the real line with a random distribution and coloring.Our results are a natural extension to a geometric context of the work initiated by Balachandran et al. (Discrete Mathematics, 2018) on arbitrary systems of unbiased representatives.

The VC-dimension of axis-parallel boxes on the Torus

We show in this paper that the VC-dimension of the family of d-dimensional axis-parallel boxes and cubes on the d-dimensional torus are both asymptotically dlog2⁡d. This is especially surprising as in most other examples the VC-dimension usually grows linearly with d in similar settings.

A generalized Faulhaber inequality, improved bracketing covers, and applications to discrepancy

We prove a generalized Faulhaber inequality to bound the sums of the j j -th powers of the first n n (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of d d -dimensional axis-parallel boxes anchored in 0 0 (or, put differently, of lower left orthants intersected with the d d -dimensional unit cube [ 0 , 1 ] d [0,1]^d ). We use these improved bracketing numbers to establish new bounds for the star-discrepancy of negatively dependent random point sets and its expectation. We apply our findings also to the weighted star-discrepancy.

Empty Axis-Parallel Boxes

Abstract We show that, for every set of $n$ points in the $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $\Omega (d/n)$ as $n\to \infty $ and $d$ is fixed. In the opposite direction, we give a construction without an empty axis-parallel box of volume $O(d^2\log d/n)$. These improve on the previous best bounds of $\Omega (\log d/n)$ and $O(2^{7d}/n)$, respectively.

Turán-type results for intersection graphs of boxes

AbstractIn this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in $${{\mathbb{R}}^d}$$ such that G contains no copy of Kt,t, then G has at most ctn( log n)2d+3 edges, where c = c(d)&gt;0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.

Discrete quantitative Helly-type theorems with boxes

Research on Helly-type theorems in combinatorial convex geometry has produced volumetric versions of Helly's theorem using witness sets and quantitative extensions of Doignon's theorem. This paper combines these philosophies and presents quantitative Helly-type theorems for the integer lattice with axis-parallel boxes as witness sets. Our main result shows that, while quantitative Helly numbers for the integer lattice grow polynomially in each fixed dimension, their variants with boxes as witness sets are uniformly bounded. We prove several colorful and fractional variations on this theorem. We also prove that the Helly number for A×A⊆R2 need not be finite even when A⊆Z is a syndetic set.

Geodesic spanners for points in [formula omitted] amid axis-parallel boxes

Let P be a set of n points in R3 amid a bounded number of obstacles. Consider the metric space M=(P,dM) where dM(p,q) is the geodesic distance of p and q, i.e., the length of a shortest path from p to q avoiding obstacles. When obstacles are axis-parallel boxes, we prove that M admits an 83-spanner with O(nlog3⁡n) edges. In other words, let S be a complete graph on n vertices where each node corresponds to a point p∈P. For nodes u and v of S corresponding to p,q∈P, the edge (u,v) is associated with the geodesic distance of p and q as its weight. We indeed prove that S admits a near linear-size t-spanner for some constant t.

On Wegner’s inequality for axis-parallel rectangles

According to an old conjecture of Wegner, the piercing number of a set of axis-parallel rectangles in the plane is at most twice the independence number (or matching number) minus 1, that is, τ(F)≤2ν(F)−1. On the other hand, the current best upper bound, due to Corea et al. (2015), is a Ologlogν(F)2 factor away from the current best lower bound. From the other direction, lower bound constructions with τ(F)≥2ν(F)−4 are known. Here we exhibit families of rectangles with τ=7 and ν=4 and thereby show that Wegner’s inequality, if true, cannot be improved for ν=4. The analogous result for ν=3, due to Wegner, dates back to 1968.A key element in our proof is establishing a connection with the Maximum Empty Box problem: Given a set P of n points inside an axis-parallel box U in Rd, find a maximum-volume axis-parallel box that is contained in U but contains no points of P in its interior.Whereas our construction can be extended to any larger independence number (ν=5,6,…), its analysis remains open.

Expected dispersion of uniformly distributed points

The dispersion of a point set in [0,1]d is the volume of the largest axis parallel box inside the unit cube that does not intersect the point set. We study the expected dispersion with respect to a random set of n points determined by an i.i.d. sequence of uniformly distributed random variables. Depending on the number of points n and the dimension d we provide an upper and a lower bound of the expected dispersion. In particular, we show that the minimal number of points required to achieve an expected dispersion less than ε∈(0,1) depends linearly on the dimension d.

Computational Geometry Column 69

We revisit the following problem called Maximum Empty Box: Given a set S of n points inside an axis-parallel box U in Rd, nd a maximum-volume axis-parallel box that is contained in U but contains no points of S in its interior.

Finding pairwise intersections of rectangles in a query rectangle

We consider the following problem: Preprocess a set S of n axis-parallel boxes in Rd so that given a query with an axis-parallel box in Rd, the pairs of boxes of S whose intersection intersects the query box can be reported efficiently. For the case that d=2, we present a data structure of size O(nlog⁡n) supporting O(log⁡n+k) query time, where k is the size of the output. This improves the previously best known result by de Berg et al. which requires O(log⁡n+klog⁡n) query time using O(nlog⁡n) space. There has been no result known for this problem for higher dimensions, except that for d=3, the best known data structure supports O(nlog2⁡n+klog2⁡n) query time using O(nnlog⁡n) space. For a fixed dimension d>2, we present a data structure supporting O(n1−δlogd−1⁡n+klogd−1⁡n) query time for any constant 0<δ<1. The size of the data structure is O(nδd−2δ+1log⁡n).