Load balancing algorithms based on gradient methods and their analysis through algebraic graph theory

The main results of this paper are based on the idea that most load balancing algorithms can be described in the framework of optimization theory. It enables to involve classical results linked with convergence, its speed and other elements. We emphasize that these classical results have been found independently and till now this connection has not been shown clearly. In this paper, we analyze the load balancing algorithm based on the steepest descent algorithm. The analysis shows that the speed of convergence is determined by eigenvalues of the Laplacian for the graph of a given load balancing system. This consideration also leads to the problems of choosing an optimal structure for a load balancing system. We prove that these optimal graphs have special Laplacians: the multiplicities of their minimal and maximal positive eigenvalues must be greater than one. Such a property is essential for strongly regular graphs, investigated in algebraic graph theory.