Approximate L(δ₁,δ₂,…,δ_t)‐coloring of trees and interval graphs

Abstract

Given a vector (δ 1 , δ 2 , . . . , δ t ) of nonincreasing positive integers, and an undirected graph G = (V , E ), an L(δ 1 , δ 2 , . . . , δ t )-coloring of G is a function f from the vertex set V to a set of nonnegative integers such thatis the distance (i.e., the minimum number of edges) between the vertices u and v . An optimal L(δ 1 , δ 2 , . . . , δ t )-coloring for G is one minimizing the largest integer used over all such colorings. Such a coloring problem has relevant applications in channel assignment for interference avoidance in wireless networks. This article presents efficient approximation algorithms for L(δ 1 , δ 2 , . . . , δ t )-coloring of two relevant classes of graphs-trees, and interval graphs. Specifically, based on the notion of strongly simplicial vertices, O(n(t +δ 1 )) and O(nt 2 δ 1 ) time algorithms are proposed to find α-approximate colorings on interval graphs and trees, respectively, where n is the number of vertices and α is a constant depending on t and δ 1 , . . . , δ t . Moreover, an O(n) time algorithm is given for the L(δ 1 , δ 2 )-coloring of unit interval graphs, which provides a 3-approximation.

Channel Assignment in Wireless Local Area Networks

“…Such a result is based on the fact that U can be computed in O(nt) time [2] and that, between two consecutive assignments of the same color c to two intervals, O(t) time is spent for at most t + 1 insertions and extractions of c through the palettes and O(δ 1 ) time is paid for updating the taboo counters. All the details can be found in [6].…”

“…A linear time algorithm has been given in [6], which colors each vertex v in O(1) time as follows. If δ 1 > 2δ 2 , then assign to vertex v the color:…”

“…, δ t )-coloring problem is usually addressed by means of heuristic approaches, like genetic algorithms, taboo search, saturation degree, simulated annealing, and ants heuristics, just to name a few [1]. However, approximation algorithms have been proposed in [6] for trees and interval graphs.…”

“…In the rest of this chapter, several coloring algorithms, originally presented in [6,7,9,19,26], are reviewed. First, some preliminary results useful to derive upper and lower bounds on the largest color needed are summarized in Section 0.2, where also the notions of t-simplicial and strongly-simplicial vertices are recalled.…”

On irreducible no‐hole L(2, 1)‐coloring of trees

“…Weighted colorings also arise in this context [20]. Algorithms to efficiently assign channels are a related topic of study (see, for example, [2], where the focus is on trees and interval graphs).…”