1-local 33/24-competitive Algorithm for Multicoloring Hexagonal Graphs

Rafal Witkowski,Janez Žerovnik

doi:10.46298/dmtcs.614

Abstract

Graph Theory In the frequency allocation problem, we are given a cellular telephone network whose geographical coverage area is divided into cells, where phone calls are serviced by assigned frequencies, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e. minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph, where each vertex knows its position in the graph. We present a 1-local 33/24-competitive distributed algorithm for multicoloring a hexagonal graph, thereby improving the previous 1-local 7/5-competitive algorithm.

Highlights

A vertex weighted graph is a triple G(E, V, d), where V is the set of vertices, E is the set of edges and d : V → N is a weight function assigning integer demands to vertices of G
The minimum number of colors needed for a proper multicoloring of G, χm(G), is called the multichromatic number
The main result of this paper is the following: Theorem 1.1 There is a 1-local distributed approximation algorithm for multicoloring hexagonal graphs which uses at most

Summary

Introduction

A vertex weighted graph is a triple G(E, V, d), where V is the set of vertices, E is the set of edges and d : V → N is a weight function assigning (non-negative) integer demands to vertices of G. Called k-local algorithms for multicoloring hexagonal graphs are studied. We will only consider the offline version as we develop an algorithm that multicolors a hexagonal graph with known demands at vertices. Our improvement is based on the idea of borrowing some colors used in the first stage and using them for the demands of the second stage (see [15]) This in particular implies that the second stage of our algorithm cannot be applied as a stand-alone algorithm for multicoloring arbitrary triangle-free hexagonal graphs. The main result of this paper is the following: Theorem 1.1 There is a 1-local distributed approximation algorithm for multicoloring hexagonal graphs which uses at most colors.

Basic definition and useful facts

Algorithm

Correctness proof

Findings

Conclusion