Graph theoretical methods for the design of parallel algorithms

Abstract

We give an overview on graph decomposition techniques to obtain fast algorithms for optimization problems on graphs.

Based on the observation that most algorithmical problems can be solved easily on trees these methods try to represent a given graph G as a tree of subgraphs of G, its components. One aims to keep the size of these components and the intersection between different components small. When using such a decomposition to solve an optimization problem on G partial solutions on different components have to be combined efficiently to obtain a global solution for G. Logical and algebraic characterization of optimization problems have been developed for this purpose. That way, algorithmic solution strategies on decomposed graphs can be derived in a uniform way for various optimization problems. They turn out to be very efficient for classes of graphs that are highly decomposable.

Furthermore extending this approach slightly, fast parallel algorithms can be designed for these problems in a very similar and general way. For this purpose we first have to discuss parallel procedures for graph decomposition. Based on such a decomposition local solutions are computed in parallel and combined to a global one. It turns out that a parallel strategy for the combining process does not have to exploit specific properties of the optimization problems to be solved. The parallel evaluation of the decomposition tree can be formalized by a transitive operation on an appropriate state space of components. Even unbalanced decomposition trees can be evaluated fast by parallel tree contraction techniques.