Model-checking problems as a basis for parameterized intractability

Most parameterized complexity classes are defined in terms of a parameterized version of the Boolean satisfiability problem (the so-called weighted satisfiability problem. For example, Downey and Fellow's W-hierarchy is of this form. But there are also classes, for example, the A-hierarchy, that are more naturally characterised in terms of model-checking problems for fragments of first-order logic. R. G. Downey et al. (1998) were the first to establish a connection between the two formalisms by giving a characterisation of the W-hierarchy in terms of first-order model-checking problems. We improve their result and then prove a similar correspondence between weighted satisfiability and model-checking problems for the A-hierarchy and the W-hierarchy. Thus we obtain very uniform characterisations of many of the most important parameterized complexity classes in both formalisms. Our results can be used to give new, simple proofs of some of the core results of structural parameterized complexity theory.