The correlation between the complexities of the nonhierarchical and hierarchical versions of graph problems

Abstract

In [Le 82, Le 87, LW 87, LW 88a, Le 89] a hierarchical graph model is discussed that allows exploitation of the hierarchical description of the graphs for the efficient solution of graph problems. The model is motivated by applications in CAD, and is based on a special form of a graph grammar. The above references contain polynomial time solutions for the hierarchical versions of many classical graph problems. However, there are also graph problems that cannot benefit from the succinctness of the hierarchical description of the graphs. In this paper we investigate whether the complexity of the hierarchical version of a graph problem can be predicted from the complexity of its nonhierarchical version. We find that the correlation between the complexities of the two versions of a graph problem is very loose; i.e., such a prediction is not possible in general. This is in contrast with corresponding results about other models of succinct graph description [PY 86, Wa 86]. Among others we resolve the complexities of the hierarchical versions of such natural graph problems as graph accessibility, clique, independent set, Hamiltonian circuit, colorability, circuit value, and network flow.