2-local 7/6-competitive algorithm for multicolouring a sub-class of hexagonal graphs

Abstract

An important optimization problem in the design of cellular networks is to assign sets of frequencies to transmitters to avoid unacceptable interference. A cellular network is often modelled as a subgraph of the infinite triangular lattice. The distributed frequency assignment problem can be abstracted as a multicolouring problem on a weighted hexagonal graph, where the weight vector represents the number of calls to be assigned at vertices. In this paper, we present a 2-local distributed algorithm for multicolouring triangle-free hexagonal graphs with no adjacent centres (i.e. vertices which has at least two neighbours in G, which are not on the same line) and with arbitrary demands. The algorithm is using only the local clique numbers at each vertex v of the given hexagonal graph, which can be computed from local information available at the vertex. We prove that the algorithm uses no more than ⌈ 7ω(G)/6⌉+5 colours for any triangle-free hexagonal graph with no adjacent centres G, without explicitly computing the global clique number ω(G). Hence the competitive ratio of the algorithm is 7/6.