New Spectral Lower Bounds on the Bisection Width of Graphs

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 λ2|V|/4 for the bisection width of a graph is well-known. The bisection width is equal to λ2|V|/4 iff all vertices are incident to λ2/2 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 λ2β|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 graphs. We use them to establish tighter lower bounds for the bisection width of 3- and 4-regular Ramanujan graphs.

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