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 onto the cube {(x1,x2,…,xd):xi∈{1,…,n},1≤i≤d} so that two vertices are adjacent if and only if their images lay on a line parallel to one of the cube edges and their distance is less than a given range parameter ω. In this paper we study large independent sets in LoS networks. We prove that the computational problem of finding a maximum 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-complete for any fixed ω≥3. In addition, we show that the problem is APX-hard when ω=n for d≥3. On the positive side, we show that LoS networks generalize chordal graphs. This implies that there exists a simple d-approximation algorithm for the maximum independent set problem in LoS networks. Finally, we describe a polynomial time approximation scheme for the maximum independent set problem in LoS networks for the case when ω is a constant and present an improved heuristic algorithm for the problem in the case ω=n.