A heuristic algorithm for small separators in arbitrary graphs

Reviewer: Eugene D. Denman

Plaisted describes an algorithm for decomposing (cutting) a graph G into subgraphs, G i , in a near-optimum manner. The resulting graphs G i have a near-minimum number of interconnections. Such an algorithm is valuable in designing multilayer integrated circuits (VLSI) as well as in building databases. In the former, the number of interconnections between layers should be minimized, and in the latter the number of branches between subsets of data should be a minimum. This type of problem is referred to as the small separator problem, where a separator problem is defined by the author as follows. Given a graph G , a cut in an n -vertex graph G =( V,E ) is a partition W, W of V . An edge separator in an n -vertex graph G =( V,E ) is a partition W, W of V such that W n3 and W n3 , where A is the number of elements in a set A . Such a separator is called small if the number of edges ( v,w ) with vW and w W is small. The algorithm described is based on the single commodity problem; it solves O n 2 log n single commodity problems and runs in O n 4 log 2 n time, with each problem requiring O n 3 time. The author describes the algorithm and presents theorems on separators, the bounds on the number of vertices in a cut set, and the probability associated with large and small cuts. He claims that although the algorithm does not come as close to finding optimal cuts as other algorithms, its worst-case time bound is better.