Fixed-parameter tractability, definability, and model checking
<|reference_start|>Fixed-parameter tractability, definability, and model checking: In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory.   We propose to consider parameterized model-checking problems for various fragments of first-order logic as generic parameterized problems and show how this approach can be useful in studying both fixed-parameter tractability and intractability. For example, we establish the equivalence between the model-checking for existential first-order logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tractability result shows that model-checking for first-order formulas is fixed-parameter tractable when restricted to a class of input structures with an excluded minor. On the intractability side, for every t >= 0 we prove an equivalence between model-checking for first-order formulas with t quantifier alternations and the parameterized halting problem for alternating Turing machines with t alternations. We discuss the close connection between this alternation hierarchy and Downey and Fellows' W-hierarchy.   On a more abstract level, we consider two forms of definability, called Fagin definability and slicewise definability, that are appropriate for describing parameterized problems. We give a characterization of the class FPT of all fixed-parameter tractable problems in terms of slicewise definability in finite variable least fixed-point logic, which is reminiscent of the Immerman-Vardi Theorem characterizing the class PTIME in terms of definability in least fixed-point logic.<|reference_end|>
@article{flum1999fixed-parameter,
    title={Fixed-parameter tractability, definability, and model checking},
    author={Joerg Flum and Martin Grohe},
    journal={arXiv preprint arXiv:cs/9910001},
    year={1999},
    archivePrefix={arXiv},
    eprint={cs/9910001},
    primaryClass={cs.CC cs.LO}
}