New spectral lower bounds on the bisection width of graphs

Abstract

The communication overhead is a major bottleneck for the execution of a process graph on a parallel computer system. In the case of two processors, the minimization of the communication can be modeled using the graph bisection problem. The spectral lower bound of λ 2| V|/4 for the bisection width of a graph is widely known. The bisection width is equal to λ 2| V|/4 iff all vertices are incident to λ 2/2 cut edges in every optimal bisection. We present a new method of obtaining tighter lower bounds on the bisection width. This method makes use of the level structure defined by the bisection. We define some global expansion properties and we show that the spectral lower bound increases with this global expansion. Under certain conditions we obtain a lower bound depending on λ 2 β | V| with 1 2 ⩽β<1 . We also present examples of graphs for which our new bounds are tight up to a constant factor. As a by-product, we derive new lower bounds for the bisection widths of 3- and 4-regular Ramanujan graphs.