Open AccessProceedings Article
The Within-Strip Discrete Unit Disk Cover Problem.
Robert Fraser,Alejandro López-Ortiz +1 more
- pp 53-58
Reads0
Chats0
TLDR
It is proved that the Within-Strip Discrete Unit Disk Cover (WSDUDC) problem is NP-complete on strips of any xed height, which is the most interesting result from a theoretical standpoint.Abstract:
We present a study of the Within-Strip Discrete Unit Disk Cover (WSDUDC) problem, which is a restricted version of the Discrete Unit Disk Cover (DUDC) problem. For the WSDUDC problem, there exists a set of points and a set of unit disks in the plane, and the points and disk centres are conned to a strip of xed height. An optimal solution to the WSDUDC problem is a set of disks of minimum cardinality that covers all points in the input set. We describe a range of approximation algorithms for the problem, including 4- and 3-approximate algorithms which apply for strips of height 2 p 2=3 0:94 and 0:8 units respectively, as well as a general scheme for any strip with less than unit height. We prove that the WSDUDC problem is NP-complete on strips of any xed height, which is our most interesting result from a theoretical standpoint. The result is also quite surprising, since a number of similar problems are tractable on strips of xed height. Finally, we discuss how these results may be applied to known DUDC approximation algorithms.read more
Citations
More filters
Journal ArticleDOI
Unit disk cover problem in 2D
TL;DR: The solution of the discrete unit disk cover problem is based on a polynomial time approximation scheme (PTAS) for the subproblem line separable discrete unit disks cover, where all the points in P are on one side of a line and covered by the union of the disks centered on the other side of that line.
Book ChapterDOI
Unit Disk Cover Problem in 2D
TL;DR: In this paper, the discrete unit disk cover problem and the rectangular region cover problem are considered as follows.
Journal ArticleDOI
Following a curve with the discrete Fréchet distance
Tim Wylie,Binhai Zhu +1 more
TL;DR: It is proved that three of the variations of the set-chain matching problem are NP-complete and an O ( | P | | S | ) time greedy solution is given for the version of the problem that is polynomial.
Journal ArticleDOI
Approximation algorithms for a variant of discrete piercing set problem for unit disks
TL;DR: This paper proposes a very simple algorithm that produces 12-approximation result in O(n log n) time, and proposes a PTAS for the same problem that can produce a solution with performance ratio in nO(k) time.
References
More filters
Journal ArticleDOI
A Greedy Heuristic for the Set-Covering Problem
TL;DR: It turns out that the ratio between the two grows at most logarithmically in the largest column sum of A when all the components of cT are the same, which reduces to a theorem established previously by Johnson and Lovasz.
Journal ArticleDOI
The Rectilinear Steiner Tree Problem is $NP$-Complete
TL;DR: The problem of determining the minimum length of an optimum rectilinear Steiner tree for a set A of points in the plane is shown to be NP-complete and the emphasis of the literature on heuristics and special case algorithms is well justified.
Journal ArticleDOI
The NP-completeness column: An ongoing guide
TL;DR: This is the fourteenth edition of a quarterly column that provides continuing coverage of new developments in the theory of NP-completeness, and readers who have results they would like mentioned (NP-hardness, PSPACE- hardness, polynomialtime-solvability, etc.), or open problems they wouldlike publicized, should send them to David S. Johnson.
Journal ArticleDOI
Approximation schemes for covering and packing problems in image processing and VLSI
Dorit S. Hochbaum,Wolfgang Maass +1 more
TL;DR: The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems and how it varies with problem parameters is illustrated.
Journal ArticleDOI
Planar Formulae and Their Uses
TL;DR: Using these results, it is able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness forPlanar generalized geography.