2,775 research outputs found

    Decomposition of multiple packings with subquadratic union complexity

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    Suppose kk is a positive integer and X\mathcal{X} is a kk-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most kk sets. Suppose there is a function f(n)=o(n2)f(n)=o(n^2) with the property that any nn members of X\mathcal{X} determine at most f(n)f(n) holes, which means that the complement of their union has at most f(n)f(n) bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that X\mathcal{X} can be decomposed into at most pp (11-fold) packings, where pp is a constant depending only on kk and ff.Comment: Small generalization of the main result, improvements in the proofs, minor correction

    Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

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    We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient (1+ε)(1+\varepsilon)-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density

    Computational Geometry Column 42

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    A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

    Constant-Factor Approximation for TSP with Disks

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    We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of nn disks in the plane, a TSP tour whose length is at most O(1)O(1) times the optimal can be computed in time that is polynomial in nn. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a O(1)O(1)-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

    Bidimensionality of Geometric Intersection Graphs

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    Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a grid minor is linear. These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes
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