Covering segments with unit squares

We study several variations of line segment covering problem with axis-parallel unit squares in $I\!\!R^2$. A set $S$ of $n$ line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at least one end-point of each segment. The variations depend on the orientation and length of the input segments. We prove some of these problems to be NP-complete, and give constant factor approximation algorithms for those problems. For some variations, we have polynomial time exact algorithms. For the general version of the problem, where the segments are of arbitrary length and orientation, and the squares are given as input, we propose a factor 16 approximation result based on multilevel linear programming relaxation technique, which may be useful for solving some other problems. Further, we show that our problems have connections with the problems studied by Arkin et al. 2015 on conflict-free covering problem. Our NP-completeness results hold for more simplified types of objects than those of Arkin et al. 2015

A Spinning Anti-de Sitter Wormhole

We construct a 2+1 dimensional spacetime of constant curvature whose spatial topology is that of a torus with one asymptotic region attached. It is also a black hole whose event horizon spins with respect to infinity. An observer entering the hole necessarily ends up at a "singularity"; there are no inner horizons. In the construction we take the quotient of 2+1 dimensional anti-de Sitter space by a discrete group Gamma. A key part of the analysis proceeds by studying the action of Gamma on the boundary of the spacetime.Comment: Latex, 28 pages, 7 postscript figures included in text, a Latex file without figures can be found at http://vanosf.physto.se/~stefan/spinning.html Replaced with journal version, minor change

A fast 25/6-approximation for the minimum unit disk cover problem

Given a point set P in 2D, the problem of finding the smallest set of unit disks that cover all of P is NP-hard. We present a simple algorithm for this problem with an approximation factor of 25/6 in the Euclidean norm and 2 in the max norm, by restricting the disk centers to lie on parallel lines. The run time and space of this algorithm is O(n log n) and O(n) respectively. This algorithm extends to any Lp norm and is asymptotically faster than known alternative approximation algorithms for the same approximation factor.Comment: 5 pages, 4 figure

Density not realizable as the Jacobian determinant of a bilipschitz map

Are every two separated nets in the plane bilipschitz equivalent? In the late 1990s, Burago and Kleiner and, independently, McMullen resolved this beautiful question negatively. Both solutions are based on a construction of a density function that is not realizable as the Jacobian determinant of a bilipschitz map. McMullen's construction is simpler than the Burago-Kleiner one, and we provide a full proof of its nonrealizability, which has not been available in the literature.Comment: 15 pages, the proof section (section 3) was significantly extended, two parts left to intuitive reasoning before were described rigorously, a small error in a constant factor in the definition of Omega was corrected (influences only calculations at the end, not a validity of the result), 1 picture adde

A PTAS for the Weighted Unit Disk Cover Problem

We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (\UDC) problem asks for a subset of disks of minimum total weight that covers all given points. \UDC\ is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted \UDC\ problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted \UDC\ problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for \UDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.Comment: We fixed several typos in this version. 37 pages. 15 figure

Minimum Covering with Travel Cost

Given a polygon and a visibility range, the Myopic Watchman Problem with Discrete Vision (MWPDV) asks for a closed path P and a set of scan points S, such that (i) every point of the polygon is within visibility range of a scan point; and (ii) path length plus weighted sum of scan number along the tour is minimized. Alternatively, the bicriteria problem (ii') aims at minimizing both scan number and tour length. We consider both lawn mowing (in which tour and scan points may leave P) and milling (in which tour, scan points and visibility must stay within P) variants for the MWPDV; even for simple special cases, these problems are NP-hard. We show that this problem is NP-hard, even for the special cases of rectilinear polygons and L_\infty scan range 1, and negligible small travel cost or negligible travel cost. For rectilinear MWPDV milling in grid polygons we present a 2.5-approximation with unit scan range; this holds for the bicriteria version, thus for any linear combination of travel cost and scan cost. For grid polygons and circular unit scan range, we describe a bicriteria 4-approximation. These results serve as stepping stones for the general case of circular scans with scan radius r and arbitrary polygons of feature size a, for which we extend the underlying ideas to a pi(r/a}+(r+1)/2) bicriteria approximation algorithm. Finally, we describe approximation schemes for MWPDV lawn mowing and milling of grid polygons, for fixed ratio between scan cost and travel cost.Comment: 17 pages, 12 figures; extended abstract appears in ISAAC 2009, full version to appear in Journal of Combinatorial Optimizatio

Divergence in right-angled Coxeter groups

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex

Deformation concentration for martensitic microstructures in the limit of low volume fraction

We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of $\Gamma$-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for $SBV^p$ functions whose jump sets have a prescribed orientation

Dimension of locally and asymptotically self-similar spaces

We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the topological dimension of the subspace. As an application of the first result, we prove the Gromov conjecture that the asymptotic dimension of every hyperbolic group G equals the topological dimension of its boundary at infinity plus 1, asdim G=dim(dG)+1. As an application of the second result, we construct Pontryagin surfaces for the asymptotic dimension, in particular, those are first examples of metric spaces X, Y with asdim(X x Y)<asdim X+asdim Y. Other applications are also given.Comment: 29 pages; this is an essentially extended and improved version of our paper `Capacity dimension of locally self-similar spaces

The Caratheodory Topology for Multiply Connected Domains II

We continue our exposition concerning the Caratheodory topology for multiply connected domains by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.Comment: 28 Pages, 3 Figure