Packing circles in circles and applications

Abstract

Packing circles in a circle is a classic mathematical problem, which has been studied for a long time. Recently, this problem received a great deal of applications in the study of wireless networks. Those applications are related to many combinatorial optimization problems raised in wireless networks. In analyzing approximation algorithms for those combinatorial optimization problems, it is required to establish an upper bound for the number of untouched unit circles which can be packed into a circle of radius r, where a unit circle is a circle with diameter one. This bound can be easily derived from some classic bounds for the number of unit circles packed into a circle. Moreover, analysis of those approximation algorithms also promoted some new problems about packing circles in circles. In this chapter, a state of the art on the developments in this research direction is presented.