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Approximation Algorithms for Maximum Independent Set of a Unit Disk Graph ∗ Gautam K

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Approximation Algorithms for Maximum Independent Set of a Unit Disk Graph ∗ Gautam K Information Processing Letters 115 (2015) 439–446 Contents lists available at ScienceDirect Information Processing Letters www.elsevier.com/locate/ipl Approximation algorithms for maximum independent set of a unit disk graph ∗ Gautam K. Das a, Minati De b, Sudeshna Kolay c, Subhas C. Nandy d, , Susmita Sur-Kolay d a Dept. of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India b Dept. of Computer Science, The Technion—Israel Institute of Technology, Israel c The Institute of Mathematical Sciences, Chennai, India d Indian Statistical Institute, Kolkata, India a r t i c l e i n f o a b s t r a c t Article history: We propose a 2-approximation algorithm for the maximum independent set problem for a 3 2 Received 21 November 2013 unit disk graph. The time and space complexities are O (n ) and O (n ), respectively. For a Received in revised form 5 November 2014 penny graph, our proposed 2-approximation algorithm works in O (n log n) time using O (n) Accepted 6 November 2014 space. We also propose a polynomial-time approximation scheme (PTAS) for the maximum Available online 15 November 2014 independent set problem for a unit disk graph. Given an integer k > 1, it produces a Communicated by R. Uehara solution of size 1 |OPT| in O (k4nσk log k + n logn) time and O (n + k logk) space, where + 1 2 (1 k ) Keywords: ≤ 7k + OPT is the subset of disks in an optimal solution and σk 3 2. For a penny graph, the Maximum independent set 1 | | 2σk + Unit disk graph proposed PTAS produces a solution of size + 1 OPT in O (2 nk n logn) time using (1 k ) Approximation algorithms O (2σk + n) space. PTAS © 2014 Elsevier B.V. All rights reserved. Computational geometry 1. Introduction graph. In this paper, we consider the problem of finding a maximum independent set (MIS) in a given unit disk Unit disk graphs play an important role in formulat- graph, where the co-ordinates (xi , yi ) of the center ci of ing several problems in mobile ad hoc networks. A unit each disk Ci are given. disk graph G = (V , E) is the intersection graph of a set The MIS problem for unit disk graph is known to be NP- 2 complete [2]. Thus, research on this topic is concentrated of circular disks C ={C1, C2, ..., Cn}, placed in R , each on designing efficient approximation algorithms. Most of having diameter 1. The center of disk Ci is denoted by ci . the related works assume that the geometric representa- Each vertex vi ∈ V corresponds to a disk Ci , and an edge tion, i.e., the layout of the set of unit disks is given. In such (vi , v j) ∈ E indicates that the corresponding pair of unit an environment, the MIS for unit disk graph is defined as disks Ci and C j intersect, i.e., δ(ci, c j) ≤ 1, where δ(a, b) is the Euclidean distance between a pair of points a, b ∈ R2. follows: In a mobile network, if all its base stations have the same C range of transmission, then these can be viewed as the Given a set of ncircular disks, each of diameter 1, placed R2 vertices of a unit disk graph. Various practical problems arbitrarily in , find a subset OPT of non-intersecting disks C on this network can be formulated in terms of a unit disk in such that OPT has the maximum cardinality among all possible subsets of non-intersecting disks in C. * Corresponding author. Previous works: A dynamic programming based shift- E-mail address: [email protected] (S.C. Nandy). ing strategy was used by Erlebach et al. [7] to design a http://dx.doi.org/10.1016/j.ipl.2014.11.002 0020-0190/© 2014 Elsevier B.V. All rights reserved. 440 G.K. Das et al. / Information Processing Letters 115 (2015) 439–446 Fig. 1. (a) A layout of unit disks, (b) corresponding unit disk graph, and (c) horizontal lines at distance 1 dividing the region into strips. polynomial-time approximation scheme (PTAS) for finding Our main results: First, we propose a 2-approximation al- a maximum weighted independent set (disjoint disks of gorithm for the MIS problem for a given unit disk graph. maximum total weight) in an intersection graph of n disks The time and space complexities of our proposed algo- 1 3 2 + 2 rithm are O (n ) and O (n ), respectively. The perspec- of arbitrary radii. They proposed a (1 k+1 ) -approxima- 4 tion algorithm1 in nO (k ) time, where k is an integer tive of this algorithm is that (i) the best known constant greater than 1. Matsui [12] showed that if the centers of factor approximation algorithm available for this problem a set of n unit disks lie inside a region bounded by a pair achieves an approximation factor 3 with time complexity 2 O (n ) [13], and (ii) by plugging r = 2in Matsui’s PTAS re- of parallel lines at a distance at most k apart, then an op- 4 √2k sult for the MIS problem of unit disk graph [12], we get 3 8 timal MIS can be obtained in O (n ) time. He also gave a 2-approximation algorithm that runs in O (n ) time us- an approximation algorithm for the MIS problem for unit 4 ing O (n ) space. Hence, our 2-approximation algorithm is 1 disk graphs, that produces a solution of size (1 − )|OPT|, r much faster than the best known 2-approximation algo- − 4 2(√r 1) 3 2r rithm for the MIS problem for unit disk graphs. and runs in O (rn ) time using O (n ) space, for any Second, we consider the MIS problem for a penny graph, positive integer r ≥ 2. Here OPT is the subset of disks in an which is a special case of unit disk graph where the cor- optimum solution. responding unit disks do not overlap, i.e., the distance The concept of thickness was introduced by van Leeu- between any two disk centers is at least 1. Here an inter- wen [11] to propose a fixed parameter tractable algorithm section between a pair of disks implies that the distance for the MIS problem for a unit disk graph. An instance of between corresponding centers is exactly 1. The MIS prob- a unit disk layout is said to have thickness τ if the re- lem for the penny graph is also NP-complete [3]. Our algo- gion containing the disks can be split into a set of strips of rithm produces a 2-approximation result for a penny graph width 1 such that each strip contains at most τ disk cen- in O (n logn) time. ters. Further, he showed that an instance of the MIS prob- 2 2τ Next, we propose a PTAS for the MIS problem for the lem with thickness τ can be solved in O (τ 2 n) time. unit disk graph. It needs to optimally solve a subproblem Agarwal et al. [1] proposed a 2-approximation algo- of the MIS problem where the centers of a set of unit rithm for the MIS problem for the rectangle intersec- disks lie inside a square region of size k × k. We show tion graph corresponding to a given set of rectangles of 3 σk log k fixed size. They also proposed a PTAS for this MIS prob- that this problem can be solved in O (k m ) time us- 1 + lem which produces a (1 + )-approximation result in ing O (k log k min(ϕk, m)) space, where m is the input k 7k k 2 2k−1 size of the subproblem, σk ≤ + 2 and ϕk ≤ 9( ) . We O (n logn + n ) time for any positive integer k. For a set 3 2 then apply the two level shifting strategy of Hochbaum of arbitrary squares or rectangles of bounded aspect ratio d−1 and Maass [10] to get a solution of the original MIS prob- in Rd, Chan [4] proposed a PTAS that runs in O (n1/ε ) lem, that is of size at least 1 |OPT|, in O (k4nσk log k) time and space with 0 < 1. They improved the space (1+ 1 )2 + k complexity to O (n) by sacrificing the running time to time using O (n k logk) space, where n is the input size d O (n1/ε ). Recently, Chan and Har-Peled [5] addressed the of the original MIS problem. The time complexity of our MIS problem for pseudo-disks in the plane. In the un- algorithm is comparable with that of [12]; but its space + 2k weighted case, for a set of n pseudo-disks, their algorithm complexity is O (n k log k) compared to O (n ) of [12]. − | | O (1/ε2) For penny graphs, the time and space complexities of our produces a solution of size (1 ε) OPT in O (n ) 2σ σ PTAS are O (2 k nk + n logn) and O (2 k + n), respectively. time; this result is similar to Erlebach’s results [7] for n disks of arbitrary radii. For the weighted case, they pro- 3 2. MIS problem for unit disk graphs posed an O (n ) time algorithm to produce an independent set of total weight Ω(|OPT|), where OPT is a subset of A layout of a set C of unit disks and the corresponding independent disks that produces maximum weight over unit disk graph are shown in Figs. 1(a) and 1(b), respec- all possible independent sets, provided the set of pseudo- tively.
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