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 certain fragments of first-order logic. Downey, Fellows, and Regan 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.Comment: Changes in since v2: Metadata update

A parametric analysis of the state-explosion problem in model checking

AbstractIn model checking, the state-explosion problem occurs when one checks a nonflat system, i.e., a system implicitly described as a synchronized product of elementary subsystems. In this paper, we investigate the complexity of a wide variety of model-checking problems for nonflat systems under the light of parameterized complexity, taking the number of synchronized components as a parameter. We provide precise complexity measures (in the parameterized sense) for most of the problems we investigate, and evidence that the results are robust

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 certain fragments of first-order logic. Downey, Fellows, and Regan 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

The Parameterized Complexity of Degree Constrained Editing Problems

This thesis examines degree constrained editing problems within the framework of parameterized complexity. A degree constrained editing problem takes as input a graph and a set of constraints and asks whether the graph can be altered in at most k editing steps such that the degrees of the remaining vertices are within the given constraints. Parameterized complexity gives a framework for examining problems that are traditionally considered intractable and developing efficient exact algorithms for them, or showing that it is unlikely that they have such algorithms, by introducing an additional component to the input, the parameter, which gives additional information about the structure of the problem. If the problem has an algorithm that is exponential in the parameter, but polynomial, with constant degree, in the size of the input, then it is considered to be fixed-parameter tractable. Parameterized complexity also provides an intractability framework for identifying problems that are likely to not have such an algorithm. Degree constrained editing problems provide natural parameterizations in terms of the total cost k of vertex deletions, edge deletions and edge additions allowed, and the upper bound r on the degree of the vertices remaining after editing. We define a class of degree constrained editing problems, WDCE, which generalises several well know problems, such as Degree r Deletion, Cubic Subgraph, r-Regular Subgraph, f-Factor and General Factor. We show that in general if both k and r are part of the parameter, problems in the WDCE class are fixed-parameter tractable, and if parameterized by k or r alone, the problems are intractable in a parameterized sense. We further show cases of WDCE that have polynomial time kernelizations, and in particular when all the degree constraints are a single number and the editing operations include vertex deletion and edge deletion we show that there is a kernel with at most O(kr(k + r)) vertices. If we allow vertex deletion and edge addition, we show that despite remaining fixed-parameter tractable when parameterized by k and r together, the problems are unlikely to have polynomial sized kernelizations, or polynomial time kernelizations of a certain form, under certain complexity theoretic assumptions. We also examine a more general case where given an input graph the question is whether with at most k deletions the graph can be made r-degenerate. We show that in this case the problems are intractable, even when r is a constant

Model-Checking Problems as a Basis for Parameterized Intractability

Most parameterized complexity classes are defined in terms of a parameterizedversion of the Boolean satisfiability problem (the so-called weightedsatisfiability problem). For example, Downey and Fellow's W-hierarchy is ofthis form. But there are also classes, for example, the A-hierarchy, that aremore naturally characterised in terms of model-checking problems for certainfragments of first-order logic. Downey, Fellows, and Regan were the first to establish a connection betweenthe two formalisms by giving a characterisation of the W-hierarchy in terms offirst-order model-checking problems. We improve their result and then prove asimilar correspondence between weighted satisfiability and model-checkingproblems for the A-hierarchy and the W^*-hierarchy. Thus we obtain very uniformcharacterisations of many of the most important parameterized complexityclasses in both formalisms. Our results can be used to give new, simple proofs of some of the coreresults of structural parameterized complexity theory.Comment: Changes in since v2: Metadata update

Model-checking problems as a basis for parameterized intractability

Vol. 1 (1:2) 2005, pp. 1–3