Fixed-parameter tractability, definability, and model checking

TL;DR

This paper explores the intersection of parameterized complexity and logic, establishing new results on fixed-parameter tractability and intractability of model-checking problems for various logical fragments and structures.

Contribution

It introduces a logical framework for analyzing parameterized problems, proving fixed-parameter tractability under certain structural restrictions and establishing equivalences with classical problems.

Findings

Model-checking for existential first-order logic is fixed-parameter tractable on structures with excluded minors.

Model-checking with t quantifier alternations is equivalent to the parameterized halting problem for alternating Turing machines.

Characterizes FPT via slicewise definability in finite variable least fixed-point logic.

Abstract

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.