We consider the maximum (weight) independent set problem in unit disk graphs. The high complexity of the existing polynomial-time approximation schemes motivated the development of faster constant-approximation algorithms. In this article, we present a 2.16-approximation algorithm that runs in O(n log 2 n) time and a 2approximation algorithm that runs in O(n 2 log n) time for the unweighted version of the problem. In the weighted version, the running times increase by an O(log n) factor. Our algorithms are based on a classic strip decomposition, but we improve over previous algorithms by efficiently using geometric data structures. We also propose a PTAS for the unweighted version.

In this article, we study approximation algorithms for the problem of computing minimum dominating set for a given set [Formula: see text] of [Formula: see text] unit disks in [Formula: see text]. We first present a simple [Formula: see text] time 5-factor approximation algorithm for this problem, where [Formula: see text] is the size of the output. The best known 4-factor and 3-factor approximation algorithms for the same problem run in time [Formula: see text] and [Formula: see text] respectively [M. De, G. K. Das, P. Carmi and S. C. Nandy, Approximation algorithms for a variant of discrete piercing set problem for unit disks, Int. J. of Computational Geometry and Appl., 22(6):461–477, 2013]. We show that the time complexity of the in-place 4-factor approximation algorithm for this problem can be improved to [Formula: see text]. A minor modification of this algorithm produces a [Formula: see text]-factor approximation algorithm in [Formula: see text] time. The same techniques can be applied to have a 3-factor and a [Formula: see text]-factor approximation algorithms in time [Formula: see text] and [Formula: see text] respectively. Finally, we propose a very important shifting lemma, which is of independent interest, and it helps to present [Formula: see text]-factor approximation algorithm for the same problem. It also helps to improve the time complexity of the proposed PTAS for the problem.

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In this paper, we consider a restricted covering problem, in which a convex polygon [Formula: see text] with [Formula: see text] vertices and an integer [Formula: see text] are given, the objective is to cover the entire region of [Formula: see text] using [Formula: see text] congruent disks of minimum radius [Formula: see text], centered on the boundary of [Formula: see text]. For [Formula: see text] and any [Formula: see text], we propose an [Formula: see text]-factor approximation algorithm for this problem, which runs in [Formula: see text] time. The best known approximation factor of the algorithm for the problem in the literature is 1.8841 [H. Du and Y. Xu: An approximation algorithm for [Formula: see text]-center problem on a convex polygon, J. Comb. Optim. 27(3) (2014) 504–518].