96 research outputs found
Constant-Factor Approximation for TSP with Disks
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 disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . 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
-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
We show that for any set of points moving along "simple" trajectories
(i.e., each coordinate is described with a polynomial of bounded degree) in
and any parameter , one can select a fixed non-empty
subset of the points of size , such that the Voronoi diagram of
this subset is "balanced" at any given time (i.e., it contains points
per cell). We also show that the bound 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 moving sensors so that at any given time their interference is
. 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 -net theory to kinetic environments
Fractional covers of hypergraphs with bounded multi-intersection
Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of hypergraphs. Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself. We show how this combinatorial result can be used to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is ≤k for some constant k. Moreover, we show a dual version of our main result for fractional hitting sets
On Interference Among Moving Sensors and Related Problems
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 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
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
We introduce the following variant of the VC-dimension. Given and a positive integer , we define to be the
size of the largest subset such that the projection of on
every subset of of size is the -dimensional cube. We show that
determining the largest cardinality of a set with a given
dimension is equivalent to a Tur\'an-type problem related to the total number
of cliques in a -uniform hypergraph. This allows us to beat the
Sauer--Shelah lemma for this notion of dimension. We use this to obtain several
results on -circuits, i.e., depth- circuits with top gate OR and
bottom fan-in at most :
* Tight relationship between the number of satisfying assignments of a
-CNF and the dimension of the largest projection accepted by it, thus
improving Paturi, Saks, and Zane (Comput. Complex. '00).
* Improved -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 complexity of the inner
product function and all degree- polynomials over in general.
The question of determining the complexity of IP was recently
posed by Golovnev, Kulikov, and Williams (ITCS'21)
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