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 by the graph bisection problem. The spectral lower bound of $\frac{{\lambda _2 |V|}} {4}$ for the bisection width of a graph is well-known. The bisection width is equal to $\frac{{\lambda _2 |V|}} {4}$ iff all vertices are incident to $\frac{{\lambda _2 }} {2}$ cut edges in every optimal bisection. We discuss the case for which this fact is not satisfied and present a new method to get tighter lower bounds on the bisection width. This method makes use of the level structure defined by the bisection. Under certain conditions we get a lower bound depending on $\frac{{\lambda _2 |V|}} {4}$ 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 graphs. We use them to establish tighter lower bounds for the bisection width of 3- and 4-regular Ramanujan graphs.

Supported by the German Science Foundation (DFG) Project SFB-376 and by the European Union ESPRIT LTR Project 20244 (ALCOM-IT).