scholarly journals Settling the APX-Hardness Status for Geometric Set Cover

2014 ◽  
Author(s):  
Nabil H. Mustafa ◽  
Rajiv Raman ◽  
Saurabh Ray
Keyword(s):  
Set Cover ◽  
Geometric Set Cover

scholarly journals Improved Approximation Algorithms for Geometric Set Cover

2006 ◽  
Vol 37 (1) ◽  
pp. 43-58 ◽  
Author(s):  
Kenneth L. Clarkson ◽  
Kasturi Varadarajan
Keyword(s):  
Approximation Algorithms ◽  
Set Cover ◽  
Geometric Set Cover

scholarly journals On the VC-dimension of half-spaces with respect to convex sets

2021 ◽  
Vol vol. 23, no. 3 (Combinatorics) ◽  
Author(s):  
Nicolas Grelier ◽  
Saeed Gh. Ilchi ◽  
Tillmann Miltzow ◽  
Shakhar Smorodinsky
Keyword(s):  
Pairwise Disjoint ◽  
Convex Sets ◽  
Range Query ◽  
Set Cover ◽  
Vc Dimension ◽  
Set Cover Problem ◽  
Structure Problem ◽  
Cover Problem ◽  
Geometric Set Cover ◽  
Disjoint Convex

A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.

An exact algorithm for a class of geometric set-cover problems

2021 ◽  
Vol 300 ◽  
pp. 25-35
Author(s):  
Claudio Contardo ◽  
Alain Hertz
Keyword(s):  
Exact Algorithm ◽  
Set Cover ◽  
Geometric Set Cover

scholarly journals Dynamic Geometric Set Cover, Revisited

2022 ◽  
pp. 3496-3528
Author(s):  
Timothy M. Chan ◽  
Qizheng He ◽  
Subhash Suri ◽  
Jie Xue
Keyword(s):  
Set Cover ◽  
Geometric Set Cover

scholarly journals Approximation algorithms for a geometric set cover problem

2012 ◽  
Vol 160 (7-8) ◽  
pp. 1039-1052 ◽  
Author(s):  
Valentin E. Brimkov ◽  
Andrew Leach ◽  
Jimmy Wu ◽  
Michael Mastroianni
Keyword(s):  
Approximation Algorithms ◽  
Set Cover ◽  
Set Cover Problem ◽  
Cover Problem ◽  
Geometric Set Cover
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