Discrete & Computational Geometry最新文献

We establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in (mathbb {R}^3) and (mathbb {R}^4). In dimension three this distance is equal to (frac{9}{5}), and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.

Kalai conjectured that if P is a simplicial d-polytope that has no missing faces of dimension (d-1), then the graph of P and the space of affine 2-stresses of P determine P up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if (2le ile d/2) and P is a simplicial d-polytope that has no missing faces of dimension (ge d-i+1), then the space of affine i-stresses of P determines the space of affine 1-stresses of P. We prove this conjecture for (1) k-stacked d-polytopes with (2le ile kle d/2-1), (2) d-polytopes that have no missing faces of dimension (ge d-2i+2), and (3) flag PL ((d-1))-spheres with generic embeddings (for all (2le ile d/2)). We also discuss several related results and conjectures. For instance, we show that if P is a simplicial d-polytope that has no missing faces of dimension (ge d-2i+2), then the ((i-1))-skeleton of P and the set of sign vectors of affine i-stresses of P determine the combinatorial type of P. Along the way, we establish the partition of unity of affine stresses: for any (1le ile (d-1)/2), the space of affine i-stresses of a simplicial d-polytope as well as the space of affine i-stresses of a simplicial ((d-1))-sphere (with a generic embedding) can be expressed as the sum of affine i-stress spaces of vertex stars. This is analogous to Adiprasito’s partition of unity of linear stresses for Cohen–Macaulay complexes.

Let (P) be a set of n points in (mathbb {R}^2). For a parameter (varepsilon in (0,1)), a subset (Csubseteq P) is an (varepsilon )-kernel of (P) if the projection of the convex hull of (C) approximates that of (P) within ((1-varepsilon ))-factor in every direction. The set (C) is a weak (varepsilon )-kernel of (P) if its directional width approximates that of (P) in every direction. Let (textsf{k}_{varepsilon }(P)) (resp. (textsf{k}^{textsf{w}}_{varepsilon }(P))) denote the minimum-size of an (varepsilon )-kernel (resp. weak (varepsilon )-kernel) of (P). We present an (O(ntextsf{k}_{varepsilon }(P)log n))-time algorithm for computing an (varepsilon )-kernel of (P) of size (textsf{k}_{varepsilon }(P)), and an (O(n^2log n))-time algorithm for computing a weak (varepsilon )-kernel of (P) of size (textsf{k}^{textsf{w}}_{varepsilon }(P)). We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of (varepsilon )-core, a convex polygon lying inside , prove that it is a good approximation of the optimal (varepsilon )-kernel, present an efficient algorithm for computing it, and use it to compute an (varepsilon )-kernel of small size.

Let K be a convex body (a compact convex set) in (mathbb {R}^d), that contains a copy of another body S in every possible orientation. Is it always possible to continuously move any one copy of S into another, inside K? As a stronger question, is it always possible to continuously select, for each orientation, one copy of S in that orientation? These questions were asked by Croft. We show that, in two dimensions, the stronger question always has an affirmative answer. We also show that in three dimensions the answer is negative, even for the case when S is a line segment – but that in any dimension the first question has a positive answer when S is a line segment. And we prove that, surprisingly, the answer to the first question is negative in dimensions four and higher for general S.

A family of fractal arrangements of circles is introduced for each imaginary quadratic field K. Collectively, these arrangements contain (up to an affine transformation) every set of circles in the extended complex plane with integral curvatures and Zariski dense symmetry group. When that set is a circle packing, we show how the ambient structure of our arrangement gives a geometric criterion for satisfying the asymptotic local–global principle. Connections to the class group of K are also explored. Among them is a geometric property that guarantees certain ideal classes are group generators.

Finding point configurations, that yield the maximum polarization (Chebyshev constant) is gaining interest in the field of geometric optimization. In the present article, we study the problem of unconstrained maximum polarization on compact sets. In particular, we discuss necessary conditions for local optimality, such as that a locally optimal configuration is always contained in the convex hull of the respective darkest points. Building on this, we propose two sequences of mixed-integer linear programs in order to compute lower and upper bounds on the maximal polarization, where the lower bound is constructive. Moreover, we prove the convergence of these sequences towards the maximal polarization.

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a “giant component” of vertices, with measure (1-o(1)) and polynomial diameter. Both bounds rely on spectral gaps—of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second—which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.

The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of n equal disks packed into a rectangle and integers k and h, we ask whether it is possible by changing positions of at most h disks to pack (n+k) disks. Thus the problem of packing equal disks is the special case of our problem with (n=h=0). While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for (h=0). Our main algorithmic contribution is an algorithm that solves the repacking problem in time ((h+k)^{mathcal {O}(h+k)}cdot |I|^{mathcal {O}(1)}), where |I| is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h.

Abstract

Let (mathcal {C}) be a set of n axis-aligned cubes of arbitrary sizes in ({mathbb R}^3) in general position. Let (mathcal {U}:=mathcal {U}(mathcal {C})) be their union, and let (kappa ) be the number of vertices on (partial mathcal {U}) ; (kappa ) can vary between O(1) and (Theta (n^2)) . We present a partition of (mathop {textrm{cl}}({mathbb R}^3setminus mathcal {U})) into (O(kappa log ^4 n)) axis-aligned boxes with pairwise-disjoint interiors that can be computed in (O(n log ^2 n + kappa log ^6 n)) time if the faces of (partial mathcal {U}) are pre-computed. We also show that a partition of size (O(sigma log ^4 n + kappa log ^2 n)) , where (sigma ) is the number of input cubes that appear on (partial mathcal {U}) , can be computed in (O(n log ^2 n + sigma log ^8 n + kappa log ^6 n)) time if the faces of (partial mathcal {U}) are pre-computed. The complexity and runtime bounds improve to (O(nlog n)) if all cubes in (mathcal {C}) are congruent and the faces of (partial mathcal {U}) are pre-computed. Finally, we show that if (mathcal {C}) is a set of arbitrary axis-aligned boxes in ({mathbb R}^3) , then a partition of (mathop {textrm{cl}}({mathbb R}^3setminus mathcal {U})) into (O(n^{3/2}+kappa )) boxes can be computed in time (O((n^{3/2}+kappa )log n)) , where (kappa ) is, as above, the number of vertices in (mathcal {U}(mathcal {C})) , which now can vary between O(1) and (Theta (n^3)) .

Inspired by work of Fröberg (1990), and Eagon and Reiner (1998), we define the total k-cut complex of a graph G to be the simplicial complex whose facets are the complements of independent sets of size k in G. We study the homotopy types and combinatorial properties of total cut complexes for various families of graphs, including chordal graphs, cycles, bipartite graphs, the prism (K_n times K_2), and grid graphs, using techniques from algebraic topology and discrete Morse theory.