Finding Large Independent Sets in Line of Sight Networks

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.. 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 = 2, the (decision version of) the problem becomes NP-hard for any fixed is an element of = 3 and even if. is chosen to be a function of n that is O(n(1-is an element of)) for any fixed omega > 0. In addition we show that the problem is also NP-hard when is an element of = n for d = 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. is constant.