5 research outputs found
Geometric Set Cover and Hitting Sets for Polytopes in
Suppose we are given a finite set of points in and a collection of
polytopes that are all translates of the same polytope . We
consider two problems in this paper. The first is the set cover problem where
we want to select a minimal number of polytopes from the collection
such that their union covers all input points . The second
problem that we consider is finding a hitting set for the set of polytopes
, that is, we want to select a minimal number of points from the
input points such that every given polytope is hit by at least one point.
We give the first constant-factor approximation algorithms for both problems.
We achieve this by providing an epsilon-net for translates of a polytope in
of size \bigO(\frac{1{\epsilon)
Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension
We consider the problem of finding a small hitting set in an infinite range space F=(Q,R) of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any delta>0, a set of size O(s_F(z^*_F)) that hits (1-delta)-fraction of R (with respect to a given measure) in time proportional to log(1/delta), where s_F(1/epsilon) is the size of the smallest epsilon-net the range space admits, and z^*_F is the value of the fractional optimal solution. This exponentially improves upon previous results which achieve the same approximation guarantees with running time proportional to poly(1/delta). Our assumptions hold, for instance, in the case when the range space represents the visibility regions of a polygon in the plane, giving thus a deterministic polynomial-time O(log z^*_F)-approximation algorithm for guarding (1-delta)-fraction of the area of any given simple polygon, with running time proportional to polylog(1/delta)
Discretization of Planar Geometric Cover Problems
We consider discretization of the 'geometric cover problem' in the plane:
Given a set of points in the plane and a compact planar object ,
find a minimum cardinality collection of planar translates of such that
the union of the translates in the collection contains all the points in .
We show that the geometric cover problem can be converted to a form of the
geometric set cover, which has a given finite-size collection of translates
rather than the infinite continuous solution space of the former. We propose a
reduced finite solution space that consists of distinct canonical translates
and present polynomial algorithms to find the reduce solution space for disks,
convex/non-convex polygons (including holes), and planar objects consisting of
finite Jordan curves.Comment: 16 pages, 5 figure