2-local 4/3-competitive algorithm for multicoloring 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 generally modeled as a subgraph of the infinite triangular lattice. Frequency assignment problem can be abstracted as a multicoloring problem on a weighted hexagonal graph, where the weights represent the number of calls to be assigned at vertices. In this paper we present a distributed algorithm for multicoloring hexagonal graphs using only the local clique numbers ω 1 (v) and ω 2 (v) at each vertex v of the given hexagonal graph, which can be computed from local information available at the vertex. We prove the algorithm uses no more than 4ω(G)/3 colors for any hexagonal graph G, without explicitly computing the global clique number ω(G). We also prove that our algorithm is 2-local, i.e., the computation at a vertex v ∈ G uses only information about the demands of vertices whose graph distance from v is less than or equal to 2.

Multicoloring of cannonball graphs

“…No matter how we store one layer onto another, for every hexagonal graph in a particular horizontal layer one of the well known algorithms [12,18,26] may be used. The best known approximation ratio is 4 3 ω(G ), where G is a hexagonal graph in a single layer (obviously ω(G ) ≤ ω(G)).…”

“…Altogether we get an algorithm that uses at most 2 · In many papers, e.g. [12,18,23,25], a strategy of borrowing was used. The same idea can be used for cannonball graphs.…”

“…This is a natural choice because it is well known that hexagonal cells provide a coverage with the optimal ratio of the distance between centers compared to the area covered by each cell. Such graphs are called hexagonal graphs [18,19,21]. Indeed, the model is a reasonable approximation for the rural cellular networks where the underlying graph is often nearly planar, and a popular example are the sets of benchmark problems based on the real cellular network around Philadelphia [2] (see the FAP website [29]).…”

“…McDiarmid and Reed proved in [12] that multicoloring of hexagonal graphs is NP-complete. In the last decade there were several results on upper bounds for the multichromatic number in terms of weighted clique number for hexagonal graphs, some of which also provide approximation algorithms that are fully distributed and run in constant time [8,9,10,12,13,14,16,18,19,20,15,21,23,24,25,27]. The best known approximation ratios are χ m (G) ≤ (4/3)ω(G) + O(1) in general [12,14,18] and χ m (G) ≤ (7/6)ω(G)+O(1) for triangle free hexagonal graphs [8,15,16] [12].…”

A 1-Local Asymptotic 13/9-Competitive Algorithm for Multicoloring Hexagonal Graphs

“…The best known competitive ratios for 0-, 1-, 2-and 4-local distributed algorithms for multicoloring on (general) hexagonal graphs are 3, 3/2, 4/3 and 4/3, respectively [8,17]. It is possible to do better for triangle-free hexagonal graphs.…”