Improved Algorithm for Maximum Independent Set on Unit Disk Graph

Abstract

In this paper, we present a 2-factor approximation algorithm for the maximum independent set problem on a unit disk graph, where the geometric representation of the graph has been given. We use dynamic programming and farthest point Voronoi diagram concept to achieve the desired approximation factor. Our algorithm runs in $O(n^2\log n)$ time and $O(n^2)$ space, where n is the input size. We also propose a polynomial time approximation scheme (PTAS) for the same problem. Given a positive integer k, it can produce a solution of size $\frac{1}{(1+\frac{1}{k})^2}|OPT|$ in $n^{O(k)}$ time, where |OPT| is the optimum size of the solution. The best known algorithm available in the literature runs in (i) $O(n^3)$ time and $O(n^2)$ space for 2-factor approximation, and (ii) $n^{O(k \log k)}$ time for PTAS [Das, G.K., De, M., Kolay, S., Nandy, S.C., Sur-Kolay, S.: Approximation algorithms for maximum independent set of a unit disk graph. Information Processing Letters 115(3), 439–446 (2015)].