Independent Sets in Restricted Line of Sight Networks

Abstract

Line of Sight (LoS) networks were designed to model wireless networks in settings which may contain obstacles restricting visibility of sensors. A graph $G=(V,E)$ is a 2-dimensional LoS network if it can be embedded in an $n \times k$ rectangular point set such that a pair of vertices in V are adjacent if and only if the embedded vertices are placed on the same row or column and are at a distance less than $\omega$. We study the Maximum Independent Set (MIS) problem in restricted LoS networks where k is a constant. It has been shown in the unrestricted case when $n=k$ and $n\rightarrow \infty$ that the MIS problem is NP-hard when $\omega > 2$ is fixed or when $\omega =O(n^{1-\epsilon })$ grows as a function of n for fixed $0<\epsilon <1$. In this paper we develop a dynamic programming (DP) algorithm which shows that in the restricted case the MIS problem is solvable in polynomial time for all $\omega$. We then generalise the DP algorithm to solve three additional problems which involve two versions of the Maximum Weighted Independent Set (MWIS) problem and a scheduling problem which exhibits LoS properties in one dimension. We use the initial DP algorithm to develop an efficient polynomial time approximation scheme (EPTAS) for the MIS problem in restricted LoS networks. This has important applications, as it provides a semi-online solution to a particular instance of the scheduling problem. Finally we extend the EPTAS result to the MWIS problem.