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Approximation Algorithms for the Discrete Piercing Set Problem for Unit Disks

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Approximation Algorithms for the Discrete Piercing Set Problem for Unit Disks CCCG 2011, Toronto ON, August 10{12, 2011 Approximation Algorithms for the Discrete Piercing Set Problem for Unit Disks Minati De∗y Gautam K. Dasz Subhas C. Nandy∗ Abstract Greene [13], and by Gonzalez [14]. Chan [5] proposed a PTAS for squares of arbitrary size with time com- 2 In this note, we shall consider constant factor approxi- plexity nO(1/ ). In fact, this algorithm works for any mation algorithms for a variation of the discrete pierc- collection of fat objects. Chan and Mahmood [6] con- ing set problem for unit disks. Here a set of points P sidered the problem for a set of axis-parallel rectangles is given; the objective is to choose minimum number of of fixed height (but of arbitrary width), and proposed a 2 points in P to pierce all the disks of unit radius centered PTAS with nO(1/ ) time complexity. at the points in P . We first propose a very simple al- The minimum clique cover problem for unit disk graph gorithm that produces a 14-factor approximation result also has a long history. The problem is known to be in O(n log n) time. Next, we improve the approxima- NP-hard [9], and a 3-factor approximation algorithm tion factor to 4 and then to 3. Both algorithms run in is easy to obtain [19]. Recently, Dumitrescu and Pach polynomial time. [12] proposed an O(n2) time randomized algorithm for the minimum clique cover problem with approximation 1 Introduction ratio 2.16. They also proposed a polynomial time ap- proximation scheme (PTAS) for this problem that runs 1/2 The piercing set of a set of objects S in IR2 is a set of in O(n ) time. It is an improvement on a previous 1/4 points Q such that each object in S contains at least PTAS with O(n ) running time [22]. one point in Q. Here the problem is, given the set S, Since, the disks do not satisfy the Helly's property2, the compute a piercing set of minimum size. Let us consider minimum piercing set problem for unit disks is different the intersection graph G = (V; E) of the objects in S. from the minimum clique cover problem for unit disk Its nodes V correspond to the members in S, and an graph. The minimum piercing set problem for disks has edge e = (u; v) E, for a pair of vertices u; v V a lot of applications in wireless communication where implies that the two2 objects corresponding to the nodes2 the objective is to place the base stations to cover a u and v intersect. A clique C in the graph G implies set of radio terminals (sensors) distributed in a region. that each pair of objects corresponding to the nodes The minimum piercing set problem for unit disks is also in C are intersecting. But, it does not imply that all NP-hard [3, 12]. Carmi et. al [3] proposed an approx- of them have a non-empty common intersection region. imation algorithm for this problem where the approx- In other words, a clique C in G does not imply that imation factor is 38. In particular, if the points are the objects corresponding to the members in C can be distributed below a straight line L, and the base sta- pierced by a single point. However, if S consists of a set tions (of same range) are allowed to be installed on or of axis-parallel rectangles, then the minimum piercing above L only then a 4-factor approximation algorithm set corresponds to the minimum clique cover 1 of the can be obtained provided all the points lie within an intersection graph of the members in S. unit distance from at least one base station. The minimum clique cover problem for a set of axis- In the discrete version of the minimum piercing set prob- parallel unit squares in IR2 is known to be NP-hard [17]. lem for unit disks, two sets of points P and Q are given. Hochbaum and Maass [16] proposed a PTAS for the The unit disks are centered at the points of P , and the minimum clique cover problem for a set of axis-parallel piercing points need to be chosen from Q. The objective 2 unit squares with time complexity nO(1/ ). The time is to choose minimum number of points from Q to pierce complexity was later improved to nO(1/) by Feder and all the disks in P . The problem is known to be NP-hard [18]. The first constant factor approximation result on ∗ Indian Statistical Institute, Kolkata, India this problem is proposed by Calinescu et al. [2]. It uses yVisiting Carleton University, Canada, during April 1, 2011 - August 27, 2011. [email protected] linear programming relaxation method to produce an zIndian Institute of Technology Guwahati, India 108-factor approximation result. The approximation re- 1The minimum clique cover problem for a graph G = (V; E) is partitioning the vertex set V into minimum number of subsets 2A set of object has the Helly property if each intersecting such that the subgraph induced by each subset is a clique. family has a non-empty intersection. 23d Canadian Conference on Computational Geometry, 2011 sult is then improved to 72 in [21], 38 in [3], and 22 in [8]. factor was 5(2h 1). − Finally, Das et al. [10] proposed an 18-factor approxi- We propose three methods that use almost similar type mation algorithm that runs in O(n log n+m log m+mn) approach for the discrete piercing problem with Q = P . time, where P = n and Q = m. j j j j The first one produces a 14-factor approximation result Another variation of the discrete piercing set problem in O(n log n) time. The second one produces a 4-factor for unit disks assumes Q = P . In other words, the solution in O(n9) time, and the last one produces a 3- unit disks corresponding to the points in P need to be factor solution in O(n18) time. Recall that the running pierced by choosing a minimum number points from P time of the existing algorithm for producing a 3-factor itself. In the literature, the problem is referred to as approximation solution is O(n625) [20]. Thus, our algo- the minimum dominating set problem for the unit disk rithm is a substantial improvement over the existing re- graph (or MDS problem in short). Here, an undirected sults in the literature. We can use this result to improve graph is constructed with nodes corresponding to the the approximation factor for the h-piercing problem [4] points in P . Between a pair of nodes there is an edge of constant radius disks to 3(2h 1) from 5(2h 1). if the distance between the two points is less than or − − equal to their common radius. A vertex in the graph dominates itself and all its neighbors. The objective is 2 Approximation Algorithms to choose minimum number of vertices to dominate all the vertices in the graph. We are given a set of points P , where each point cor- responds to a unit disk centered at that point. The The problem is known to be NP-hard [7]. Ambuhl objective is to choose a subset P 0 P of minimum car- et al. [1] first proposed an approximation algorithm dinality such that the disk corresponding⊆ to each point for this problem. They considered the weighted ver- in P contains at least one member of P 0. sion of the problem where each node is attached with a positive weight. The objective is to find the mini- mum weight dominating set of the nodes in the graph. 2.1 A simple 14-factor approximation algorithm The approximation factor of their proposed algorithm is 72. Huang et al. [15] improved the approximation Consider a partitioning of the plane into a grid whose factor of the same problem to 6 + . Dai and Yu [11] each cell is of size p1 p1 . Since the maximum distance 2 × 2 further improved the approximation factor to 5 + . between any two points in a grid cell is less than or equal Though they have not analyzed the time complexity of to 1, we can pierce all the disks centered at points of P n9 their proposed algorithm, their algorithm needs O( 2 ) in a particular cell by choosing any one member p P time. Recently, Zou et al. [23] proposed a polynomial lying in that cell. In other words, if we draw a disk2 of time 4+ factor approximation algorithm. Nieberg and unit radius, and centered at p, it covers all the points Hurink [20] proposed an O(nc) time PTAS for comput- lying inside that cell. Note that, it may cover point(s) ing the minimum dominating set for unit disk graphs, in the other cell(s). But, we show that a disk centered where c = (2r + 1)2, and r is an integer satisfying at a point p P inside a grid cell may cover (some or (2r + 1)2 < (1 + )r=2. It accepts any undirected graph all) points in2 at most 14 other grid cells. as input, and returns a (1 + ) factor approximation solution for the dominating set problem, or a certifi- 1 2 3 4 5 cate indicating that the input graph is not a unit disk 6 7 8 9 10 graph. For a 2-factor approximation result, the worst- case running time is obtained by setting = 1; in that 11 12 A B 14 15 case, r will be equal to 22.
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