94 research outputs found

    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

    On interference among moving sensors and related problems

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    We show that for any set of nn points moving along "simple" trajectories (i.e., each coordinate is described with a polynomial of bounded degree) in d\Re^d and any parameter 2kn2 \le k \le n, one can select a fixed non-empty subset of the points of size O(klogk)O(k \log k), such that the Voronoi diagram of this subset is "balanced" at any given time (i.e., it contains O(n/k)O(n/k) points per cell). We also show that the bound O(klogk)O(k \log k) is near optimal even for the one dimensional case in which points move linearly in time. As applications, we show that one can assign communication radii to the sensors of a network of nn moving sensors so that at any given time their interference is O(nlogn)O(\sqrt{n\log n}). We also show some results in kinetic approximate range counting and kinetic discrepancy. In order to obtain these results, we extend well-known results from ε\varepsilon-net theory to kinetic environments

    On Interference Among Moving Sensors and Related Problems

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    We show that for any set of n moving points in R^d and any parameter 2<=k<n, one can select a fixed non-empty subset of the points of size O(k log k), such that the Voronoi diagram of this subset is "balanced" at any given time (i.e., it contains O(n/k) points per cell). We also show that the bound O(k log k) is near optimal even for the one dimensional case in which points move linearly in time. As an application, we show that one can assign communication radii to the sensors of a network of nn moving sensors so that at any given time, their interference is O( (n log n)^0.5). This is optimal up to an O((log n)^0.5) factor

    The ?-t-Net Problem

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    We study a natural generalization of the classical ?-net problem (Haussler - Welzl 1987), which we call the ?-t-net problem: Given a hypergraph on n vertices and parameters t and ? ? t/n, find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least ? n contains a set in S. When t=1, this corresponds to the ?-net problem. We prove that any sufficiently large hypergraph with VC-dimension d admits an ?-t-net of size O((1+log t)d/? log 1/?). For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O(1/?)-sized ?-t-nets. We also present an explicit construction of ?-t-nets (including ?-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of ?-nets (i.e., for t=1), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest

    A Variant of the VC-Dimension with Applications to Depth-3 Circuits

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    We introduce the following variant of the VC-dimension. Given S{0,1}nS \subseteq \{0, 1\}^n and a positive integer dd, we define Ud(S)\mathbb{U}_d(S) to be the size of the largest subset I[n]I \subseteq [n] such that the projection of SS on every subset of II of size dd is the dd-dimensional cube. We show that determining the largest cardinality of a set with a given Ud\mathbb{U}_d dimension is equivalent to a Tur\'an-type problem related to the total number of cliques in a dd-uniform hypergraph. This allows us to beat the Sauer--Shelah lemma for this notion of dimension. We use this to obtain several results on Σ3k\Sigma_3^k-circuits, i.e., depth-33 circuits with top gate OR and bottom fan-in at most kk: * Tight relationship between the number of satisfying assignments of a 22-CNF and the dimension of the largest projection accepted by it, thus improving Paturi, Saks, and Zane (Comput. Complex. '00). * Improved Σ33\Sigma_3^3-circuit lower bounds for affine dispersers for sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture under which we get further improvement. * We make progress towards settling the Σ32\Sigma_3^2 complexity of the inner product function and all degree-22 polynomials over F2\mathbb{F}_2 in general. The question of determining the Σ33\Sigma_3^3 complexity of IP was recently posed by Golovnev, Kulikov, and Williams (ITCS'21)
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