Dynamic Programming Optimization in Line of Sight Networks

Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles restricting node visibility. For fixed positive integer $d$, and positive integer $ω$, a graph $G=(V,E)$ is a ($d$-dimensional) LoS network with range parameter $ω$ if it can be embedded in a cube of side size $n$ of the $d$-dimensional integer grid so that each pair of vertices in $V$ are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than $ω$. In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally for any $ω$ on {\em narrow} LoS networks (i.e. networks which can be embedded in a $n \times k \times k \ldots \times k$ region, for some fixed $k$ independent of $n$). In the unrestricted case it has been shown that the MIS problem is NP-hard when $ ω> 2$ (the hardness proof goes through for any $ω=O(n^{1-δ})$, for fixed $0<δ<1$). We describe how DP can be used as a building block in the design of good approximation algorithms. In particular we present a 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary $d$-dimensional LoS networks. Finally we comment on how the approach can be adapted to solve a number of important optimization problems in LoS networks.