A Systematic Approach to Computations on Decomposable Graphs

Abstract

In this paper, we try to build a bridge between pure theoretical approach to computations on decomposable graphs and heuristics, used in practice for treatment of particular cases of them. In theory, Feferman and Vaught in 1959 proposed a method to reduce solution of First Order definable problems on Disjoint Union of structures to solutions of derived problems on the components with some post-processing of the obtained results. In practice, the literature is very reach in examples of particular methods to deal with different variations of graphs, built from components. From the theoretical point of view we adapt and generalize the Feferman-Vaught method. We define a new kind of decomposable graphs: sum-like graphs and propose a new systematic approach, which allows us to reduce the solution of Monadic Second Order definable problems on such graphs to the solution of effectively derivable Monadic Second Order definable problems on the components. From the practical point of view, we consider in great details one application of our approach in the field of parallel computations on distributed data.