Channel Assignment with Separation for Interference Avoidance in Wireless Networks

Given an integer $\sigma > 1$, a vector $(\delta_1, \delta_2, \ldots, \delta_{\sigma-1})$ of nonnegative integers, and an undirected graph $G=(V,E)$, an $L(\delta_1, \delta_2, \ldots,\delta_{\sigma-1})$-coloring of $G$ is a function $f$ from the vertex set $V$ to a set of nonnegative integers such that $| f(u) -f(v) | \ge \delta_i$, if $d(u,v) = i, \ 1 \le i \le \sigma-1, \ $ where $d(u,v)$ is the distance (i.e. the minimum number of edges) between the vertices $u$ and $v$. An optimal $L(\delta_1, \delta_2, \ldots,\delta_{\sigma-1})$-coloring for $G$ is one using the smallest range $\lambda$ of integers over all such colorings. This problem has relevant application in channel assignment for interference avoidance in wireless networks, where channels (i.e. colors) assigned to interfering stations (i.e. vertices) at distance $i$ must be at least $\delta_i$ apart, while the same channel can be reused in vertices whose distance is at least $\sigma$. In particular, two versions of the coloring problem -- $L(2,1,1)$, and $L(\delta_1, 1, \ldots,1)$ -- are considered. Since these versions of the problem are $NP$-hard for general graphs, efficient algorithms for finding optimal colorings are provided for specific graphs modeling realistic wireless networks including rings, bidimensional grids, and cellular grids.