Discrete & Computational Geometry最新文献

We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the “longest increasing subsequence” have Ehrhart quasi-polynomials which are honest polynomials, even though they are just rational polytopes in general. We do this by defining a continuous, piecewise-linear bijection to a certain Gelfand–Tsetlin polytope. This bijection is not an integral equivalence but it respects lattice points in the appropriate way to imply that the two polytopes have the same Ehrhart (quasi-)polynomials. In fact, the bijection is essentially the Robinson–Schensted–Knuth correspondence.

We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and translations. The number of rectangles is the rank of the Dehn invariant of the polygon. The same method can also be used to dissect an axis-parallel polygon into a simple polygon with the minimum possible number of edges. When rotations or reflections are allowed, we can approximate the minimum number of rectangles to within a factor of two.

We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.

The Art Gallery problem is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow in 24th Annual European Symposium on Algorithms (Aarhus 2016)], the classic variant has an ({{mathcal {O}}}(log k))-approximation algorithm [Bonnet and Miltzow in 33rd International Symposium on Computational Geometry (Brisbane 2017)], and it may require irrational guards [Abrahamsen et al. in 33rd International Symposium on Computational Geometry (Brisbane 2017)]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be (exists {mathbb {R}})-complete [Abrahamsen et al. in 50th Annual ACM SIGACT Symposium on Theory of Computing (Los Angeles 2018)]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP. Given the first discovery, the following question was posed by Giannopoulos [Lorentz Workshop on Fixed-Parameter Computational Geometry (Leiden 2016)]: Is Art Gallery FPT with respect to r, the number of reflex vertices? In light of the developments above, we focus on the variant where G and C consist of all the vertices of P, called Vertex-Vertex Art Gallery. Apart from being a variant of Art Gallery, this case can also be viewed as the classic Dominating Set problem in the visibility graph of a polygon. In this article, we show that the answer to the question by Giannopoulos is positive: Vertex-Vertex Art Gallery is solvable in time (r^{{{mathcal {O}}}(r^2)}hspace{0.55542pt}{cdot }hspace{1.66656pt}n^{{{mathcal {O}}}(1)}). Furthermore, our approach extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT as well. To this end, we utilize structural properties of “almost convex polygons” to present a two-stage reduction from Vertex-Vertex <

A (k, m)-Furstenberg set is a subset (S subset {mathbb {F}}_q^n) with the property that each k-dimensional subspace of ({mathbb {F}}_q^n) can be translated so that it intersects S in at least m points. Ellenberg and Erman (Algebra Number Theory 10(7), 1415–1436 (2016)) proved that (k, m)-Furstenberg sets must have size at least (C_{n,k}m^{n/k}), where (C_{n,k}) is a constant depending only n and k. In this paper, we adopt the same proof strategy as Ellenberg and Erman, but use more elementary techniques than their scheme-theoretic method. By modifying certain parts of the argument, we obtain an improved bound on (C_{n,k}), and our improved bound is nearly optimal for an algebraic generalization the main combinatorial result. We also extend our analysis to give lower bounds for sets that have large intersection with shifts of a specific family of higher-degree co-dimension (n-k) varieties, instead of just co-dimension (n-k) subspaces.

Kovaleva and Spieksma (SIAM J Discrete Math 20(3):48–768, 2006) considered the problem of stabbing a given set of horizontal line segments with the smallest number of horizontal and vertical lines. The standard LP relaxation for this problem is easily shown to have an integrality gap of at most 2 by treating the horizontal and vertical lines separately. However, Kovaleva and Spieksma observed that threshold rounding can be used to obtain an integrality gap of (e/(e-1) approx 1.58) which is also shown to be tight. This is one of the rare known examples where the obvious upper bound of 2 on the integrality gap of the standard LP relaxation can be improved. Our goal in this paper is to extend their proof to two other problems where the goal is to stab a set (mathcal {R}) of objects with horizontal and vertical lines: in the first problem (mathcal {R}) is a set of horizontal and vertical line segments, and in the second problem (mathcal {R}) is a set of unit sized squares. The proof of Kovaleva and Spieksma essentially shows the existence of an appropriate threshold which yields the improved approximation factor. We begin by showing that a random threshold picked from an appropriate distribution works. This reduces the problem to finding an appropriate distribution for a desired approximation ratio. In the first problem, we show that the required distribution can be found by solving a linear program. In the second problem, while it seems harder to find the optimal distribution, we show that using the uniform distribution an improved approximation factor can still be obtained by solving a number of linear programs.

Let (textbf{p}) be a configuration of n points in (mathbb R^d) for some n and some (d ge 2). Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing (textbf{p}) given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when (textbf{p}) will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has already been used for the simpler problem of reconstruction using unlabeled edge lengths. This paper also provides a complete proof that this process must work in that edge-setting when given a sufficiently rich set of edge measurements and assuming that (textbf{p}) is generic.

A lattice polytope translated by a rational vector is called an almost integral polytope. In this paper, we study Ehrhart quasi-polynomials of almost integral polytopes. We study the connection between the shape of polytopes and the algebraic properties of the Ehrhart quasi-polynomials. In particular, we prove that lattice zonotopes and centrally symmetric lattice polytopes are characterized by Ehrhart quasi-polynomials of their rational translations.

We begin a systematic study of the region of possible values of the volumes of Minkowski subset sums of a collection of M compact sets in (mathbb {R}^d), which we call the Lyusternik region, and make some first steps towards describing it. Our main result is that a fractional generalization of the Brunn–Minkowski–Lyusternik inequality conjectured by Bobkov et al. (in: Houdré et al. (eds) Concentration, functional inequalities and isoperimetry. Contemporary mathematics, American Mathematical Society, Providence, 2011) holds in dimension 1. Even though Fradelizi et al. (C R Acad Sci Paris Sér I Math 354(2):185–189, 2016) showed that it fails in general dimension, we show that a variant does hold in any dimension.

A grounded L-graph is the intersection graph of a collection of “L” shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number (omega ) has chromatic number at most (17omega ^4). This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially (chi )-bounded. We also survey (chi )-boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.