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.