D-dimensional Boxes Research Articles

Sidon–Ramsey and $$B_{h}$$-Ramsey numbers

AbstractFor a given positive integer k, the Sidon–Ramsey number $${{\,\textrm{SR}\,}}(k)$$ SR ( k ) is defined as the minimum value of n such that, in every partition of the set [1, n] into k parts, there exists a part that contains two distinct pairs of numbers with the same sum, i.e., one of the parts is not a Sidon set. In this paper, we investigate the asymptotic behavior of this parameter and two generalizations of it. The first generalization involves replacing pairs of numbers with h-tuples, such that in every partition of [1, n] into k parts, there exists a part that contains two distinct h-tuples with the same sum, i.e., there is a part that is not a $$B_h$$ B h set. The second generalization considers the scenario where the interval [1, n] is substituted with a d-dimensional box of the form $$\prod _{i=1}^d[1,n_i]$$ ∏ i = 1 d [ 1 , n i ] . For the general case of $$h\ge 3$$ h ≥ 3 and d-dimensional boxes, before applying our method to obtain the Ramsey-type result, we establish an upper bound for the corresponding density parameter.

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.

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.

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.

Computing the depth distribution of a set of boxes

Motivated by the analysis of range queries in databases, we introduce the computation of the depth distribution of a set B of n d-dimensional boxes (i.e., axis aligned d-dimensional hyperrectangles), which generalizes the computation of the Klee's measure and maximum depth of B. We present an algorithm to compute the depth distribution running in time within O(nd+12log⁡n), using space within O(nlog⁡n), and refine these upper bound for various measures of difficulty of the input instances. Moreover, we introduce conditional lower bounds for this problem which not only provide insights on how fast the depth distribution can be computed, but also clarify the relation between the Depth Distribution problem and other fundamental problems in computer science.

Online packing of d-dimensional boxes into the unit cube

Any sequence of d-dimensional boxes of edge length smaller than or equal to 1 with total volume not greater than (3-2sqrt{2})cdot 3^{-d} can be packed online into the d-dimensional unit cube.

Computing coverage kernels under restricted settings

Given a set B of d-dimensional boxes (i.e., axis-aligned hyperrectangles), a minimum coverage kernel is a subset of B of minimum size covering the same region as B. Computing it is NP-hard, but as for many similar NP-hard problems (e.g., Box Cover, and Orthogonal Polygon Covering), the problem becomes solvable in polynomial time under restrictions on B. We show that computing minimum coverage kernels remains NP-hard even when restricting the graph induced by the input to a highly constrained class of graphs. Alternatively, we present two polynomial-time approximation algorithms for this problem: one deterministic with an approximation ratio within O(log⁡n), and one randomized with an improved approximation ratio within O(lg⁡OPT) (with high probability).

One-dimensional Kac model of dense amorphous hard spheres

We introduce a new model of hard spheres under confinement for the study of the glass and jamming transitions. The model is a one-dimensional chain of the d-dimensional boxes each of which contains the same number of hard spheres, and the particles in the boxes of the ends of the chain are quenched at their equilibrium positions. We focus on the infinite-dimensional limit of the model and analytically compute the glass transition densities using the replica liquid theory. From the chain length dependence of the transition densities, we extract the characteristic length scales at the glass transition. The divergence of the lengths are characterized by the two exponents, for the dynamical transition and −1 for the ideal glass transition, which are consistent with those of the p-spin mean-field spin glass model. We also show that the model is useful for the study of the growing length scale at the jamming transition.

Structural Parameterizations for Boxicity

The boxicity of a graph G is the least integer d such that G has an intersection model of axis-aligned d-dimensional boxes. Boxicity, the problem of deciding whether a given graph G has boxicity at most d, is NP-complete for every fixed \(d \ge 2\). We show that Boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al. (2010), that Boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that Boxicity admits an additive 1-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. (2010) that Boxicity remains NP-complete even on graphs of constant treewidth.

Near-Linear Approximation Algorithms for Geometric Hitting Sets

Given a range space $(X,\mathcal{R})$ , where $\mathcal{R}\subset2^{X}$ , the hitting set problem is to find a smallest-cardinality subset H⊆X that intersects each set in $\mathcal{R}$ . We present near-linear-time approximation algorithms for the hitting set problem in the following geometric settings: (i) $\mathcal{R}$ is a set of planar regions with small union complexity. (ii) $\mathcal{R}$ is a set of axis-parallel d-dimensional boxes in ℝ d . In both cases X is either the entire ℝ d , or a finite set of points in ℝ d . The approximation factors yielded by the algorithm are small; they are either the same as, or within very small factors off the best factors known to be computable in polynomial time.

The number of eigenstates: counting function and heat kernel

The main aim of this paper is twofold: (1) revealing a relation between the counting function N(lambda) (the number of the eigenstates with eigenvalue smaller than a given number) and the heat kernel K(t), which is still an open problem in mathematics, and (2) introducing an approach for the calculation of N(lambda), for there is no effective method for calculating N(lambda) beyond leading order. We suggest a new expression of N(lambda) which is more suitable for practical calculations. A renormalization procedure is constructed for removing the divergences which appear when obtaining N(lambda) from a nonuniformly convergent expansion of K(t). We calculate N(lambda) for D-dimensional boxes, three-dimensional balls, and two-dimensional multiply-connected irregular regions. By the Gauss-Bonnet theorem, we generalize the simply-connected heat kernel to the multiply-connected case; this result proves Kac's conjecture on the two-dimensional multiply-connected heat kernel. The approaches for calculating eigenvalue spectra and state densities from N(lambda) are introduced.

Boxicity and maximum degree

A d-dimensional box is a Cartesian product of d closed intervals on the real line. The boxicity of a graph is the minimum dimension d such that it is representable as the intersection graph of d-dimensional boxes. We give a short constructive proof that every graph with maximum degree D has boxicity at most 2 D 2 . We also conjecture that the best upper bound is linear in D.

The largest k-ball in a d-dimensional box

This paper considers d-dimensional boxes of the form { x|− a i ≤ x i ≤ a i } in E d . It establishes a formula for the radius of the largest k-dimensional ball contained in such a d-dimensional box.

On-Line Potato-Sack Algorithm Efficient for Packing Into Small Boxes

The aim of this paper is to show an on-line algorithm for packing sequences of d-dimensional boxes of edge lengths at most 1 in a box of edge lengths at least 1. It is more efficient than a previously known algorithm in the case when packing into a box with short edges. In particular, our method permits packing every sequence of boxes of edge lengths at most 1 and of total volume at most \((1 - \frac{1} {2}\sqrt 3 )^{d - 1} \) in the unit cube. For packing sequences of convex bodies of diameters at most 1 the result is d! times smaller.

CONNECTION BETWEEN ζ AND CUTOFF REGULARIZATIONS OF CASIMIR ENERGIES

We study the connection between ζ- and cutoff-regularized Casimir energies for scalar fields. We show that, in general, both regularization schemes lead to divergent contributions, and to minimal finite parts which do not coincide. We determine the relationships among the various coefficients appearing in one approach and the other. We discuss the agreement with our predictions in the case of scalar fields in d-dimensional boxes under periodic boundary conditions. Finally, we apply our results to massless scalar fields in balls, an example where ambiguities remain under the physical prescriptions usually imposed to extract a finite result.

How many atoms can be defined by boxes?

We study the functionb(n, d), the maximal number of atoms defined byn d-dimensional boxes, i.e. parallelopipeds in thed-dimensional Euclidean space with sides parallel to the coordinate axes. We characterize extremal interval families definingb(n, 1)=2n-1 atoms and we show thatb(n, 2)=2n 2-6n+7. We prove that for everyd, $$b^* (d) = \mathop {\lim }\limits_{n \to \infty } b(n,d)/n^d $$ exists and $$1 \leqq (d/2)\sqrt[d]{{b^* (d)}} \leqq e$$ . Moreover, we obtainb*(3)=8/9.

Covering and coloring problems for relatives of intervals

The following generalizations and relatives of interval families are studied in the paper: arcs of a circle, multiple intervals, chords of a circle, d-dimensional boxes and multiple boxes. For these families we survey results and problems concerning the dependence of the transversal number on the packing number and the dependence of the coloring number on the clique number. The intersection graphs of the underlying set systems are circular arc graphs, multiple interval graphs, circle graphs (called also overlap graphs), box graphs and multiple box graphs. Thus most of our problems and results concern the relation between the clique-cover number and stability number (ϑ and α), or between the chromatic number and clique number (χ and ω) of these graphs.