Finding Large Independent Sets in Line of Sight Networks

Abstract

Line of Sight (LoS) networks provide a model of wireless communication which incorporates visibility constraints. Vertices of such networks can be embedded in finite d-dimensional grids of size n, and two vertices are adjacent if they share a line of sight and are at distance less than \(\omega \). In this paper we study large independent sets in LoS networks. We prove that the computational problem of finding a largest independent set can be solved optimally in polynomial time for one dimensional LoS networks. However, for \(d\ge 2\), the (decision version of) the problem becomes NP-hard for any fixed \(\omega \ge 3\) and even if \(\omega \) is chosen to be a function of n that is \(O(n^{1-\epsilon })\) for any fixed \(\epsilon > 0\). In addition we show that the problem is also NP-hard when \(\omega = n\) for \(d \ge 3\). This result extends earlier work which showed that the problem is solvable in polynomial time for gridline graphs when \(d=2\). Finally we describe simple algorithms that achieve constant factor approximations and present a polynomial time approximation scheme for the case where \(\omega \) is constant.