7,151 research outputs found

    An Integer Programming Formulation Using Convex Polygons for the Convex Partition Problem

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    A convex partition of a point set P in the plane is a planar partition of the convex hull of P into empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the convex hull of P and the interiors of the polygons are pairwise disjoint. Moreover, no polygon is allowed to contain a point of P in its interior. The problem is to find a convex partition with the minimum number of internal faces. The problem has been shown to be NP-hard and was recently used in the CG:SHOP Challenge 2020. We propose a new integer linear programming (IP) formulation that considerably improves over the existing one. It relies on the representation of faces as opposed to segments and points. A number of geometric properties are used to strengthen it. Data sets of 100 points are easily solved to optimality and the lower bounds provided by the model can be computed up to 300 points

    On a decomposition of regular domains into John domains with uniform constants

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    We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain ΩR2\Omega \subset {\Bbb R}^2 with C1C^1-boundary there is a corresponding partition Ω=Ω1ΩN\Omega = \Omega_1 \cup \ldots \cup \Omega_N with j=1NH1(ΩjΩ)θ\sum_{j=1}^N \mathcal{H}^1(\partial \Omega_j \setminus \partial \Omega) \le \theta such that each component is a John domain with a John constant only depending on θ\theta. The result implies that many inequalities in Sobolev spaces such as Poincar\'e's or Korn's inequality hold on the partition of Ω\Omega for uniform constants, which are independent of Ω\Omega

    On k-Convex Polygons

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    We introduce a notion of kk-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{kk-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{22-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{O(nlogn)O(n \log n)} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{22-convex} objects considered.Comment: 23 pages, 19 figure

    Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation

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    We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.Comment: 21 pages, 6 figure

    Minimum Convex Partitions and Maximum Empty Polytopes

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    Let SS be a set of nn points in Rd\mathbb{R}^d. A Steiner convex partition is a tiling of conv(S){\rm conv}(S) with empty convex bodies. For every integer dd, we show that SS admits a Steiner convex partition with at most (n1)/d\lceil (n-1)/d\rceil tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d3d\geq 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any nn points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n)\omega(1/n). Here we give a (1ε)(1-\varepsilon)-approximation algorithm for computing the maximum volume of an empty convex body amidst nn given points in the dd-dimensional unit box [0,1]d[0,1]^d.Comment: 16 pages, 4 figures; revised write-up with some running times improve
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