In this article, we consider the problem of computing minimum dominating set for a given set @math of @math points in @math . Here the objective is to find a minimum cardinality subset @math of @math such that the union of the unit radius disks centered at the points in @math covers all the points in @math . We first propose a simple 4-factor and 3-factor approximation algorithms in @math and @math time respectively improving time complexities by a factor of @math and @math respectively over the best known result available in the literature [M. De, G.K. Das, P. Carmi and S.C. Nandy, Approximation algorithms for a variant of discrete piercing set problem for unit disk , Int. J. of Comp. Geom. and Appl., to appear]. Finally, we propose a very important shifting lemma, which is of independent interest and using this lemma we propose a @math -factor approximation algorithm and a PTAS for the minimum dominating set problem.
In the discrete unit disk cover (DUDC) problem, two sets @math and @math of points in @math are given, the objective is to choose minimum number of unit disks @math centered at the points in @math such that the union of the disks in @math covers all the points in @math . Johnson @cite_7 proved that the DUDC problem is NP-hard. Mustafa and Ray in 2010 @cite_17 proposed a @math -approximation algorithm for @math (PTAS) for the DUDC problem using @math -net based local improvement approach. The fastest algorithm is obtained by setting @math for a 3-factor approximation algorithm, which runs in @math ) time, where @math and @math are number of unit radius disks and number of points respectively @cite_5 . The high complexity of the PTAS leads to further research on constant factor approximation algorithms for the DUDC problem. A series of constant factor approximation algorithms for DUDC problem are available in the literature:
@cite_5 @cite_7 @cite_17
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Given a set of n points and a set of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in is covered by at least one disk in or not and (ii) if so, then find a minimum cardinality subset such that the unit disks in cover all the points in . The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard. The general set cover problem is not approximable within , for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is . The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time . 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. Given a set @math of m unit disks and a set @math of n points in the plane, the discrete unit disk cover problem is to select a minimum cardinality subset @math to cover @math . This problem is NP-hard [14] and the best previous practical solution is a 38-approximation algorithm by [5]. We first consider the line-separable discrete unit disk cover problem (the set of disk centers can be separated from the set of points by a line) for which we present an O(n(log n + m))-time algorithm that finds an exact solution. Combining our line-separable algorithm with techniques from the algorithm of [5] results in an O(m2n4) time 22-approximate solution to the discrete unit disk cover problem. Given a polygon @math , for two points @math and @math contained in the polygon, their is the length of the shortest @math -path within @math . A of radius @math centered at a point @math is the set of points in @math whose geodesic distance to @math is at most @math . We present a polynomial time @math -approximation algorithm for finding a densest geodesic unit disk packing in @math . Allowing arbitrary radii but constraining the number of disks to be @math , we present a @math -approximation algorithm for finding a packing in @math with @math geodesic disks whose minimum radius is maximized. We then turn our focus on of @math and present a @math -approximation algorithm for covering @math with @math geodesic disks whose maximal radius is minimized. Furthermore, we show that all these problems are @math -hard in polygons with holes. Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius.