55,246 research outputs found
On the Discrete Unit Disk Cover Problem
Abstract. Given a set P of n points and a set D of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in P is covered by at least one disk in D or not and (ii) if so, then find a minimum cardinality subset D ∗ ⊆ D such that unit disks in D ∗ cover all the points in P. The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard [14]. The general set cover problem is not approx-imable within c log |P|, for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we pro-vide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is O(n log n+m logm+mn). The previ-ous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time O(m2n4).
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
Covering the Boundary of a Simple Polygon with Geodesic Unit Disks
We consider the problem of covering the boundary of a simple polygon on n
vertices using the minimum number of geodesic unit disks. We present an O(n
\log^2 n+k) time 2-approximation algorithm for finding the centers of the
disks, with k denoting the number centers found by the algorithm
- …