Mark Weyer

Bounded fixed-parameter tractability and log2n nondeterministic bits

Motivated by recent results showing that there are natural parameterized problems that are fixed-parameter tractable, but can only be solved by fixed-parameter tractable algorithms the running time of which depends nonelementarily on the parameter, we propose a notion of bounded fixed-parameter tractability, where the dependence of the running time on the parameter is restricted to be singly exponential. We develop a basic theory that is centred around the class EPT of tractable problems and an EW-hierarchy of classes of intractable problems, both in the bounded sense. By and large, this theory is similar to the established unbounded parameterized complexity theory, but there are some remarkable differences. Most notably, certain natural model-checking problems that are known to be fixed-parameter tractable in the unbounded sense have a very high complexity in the bounded theory. The problem of computing the VC-dimension of a family of sets, which is known to be complete for the class W[1] in the unbounded theory, is complete for the class EW[3] in the bounded theory. It turns out that our bounded parameterized complexity theory is closely related to the classical complexity theory of problems that can be solved by a nondeterministic polynomial time algorithm that only uses log^2n nondeterministic bits, and in particular to the classes LOGSNP and LOGNP introduced by Papadimitriou and Yannakakis.

Understanding the Complexity of Induced Subgraph Isomorphisms

We study left-hand side restrictions of the induced subgraph isomorphismproblem: Fixing a class , for given graphs G and arbitrary Hwe ask for induced subgraphs of Hisomorphic to G. For the homomorphism problem this kind of restriction has been studied by Grohe and Dalmau, Kolaitis and Vardi for the decision problem and by Dalmau and Jonsson for its counting variant. We give a dichotomy result for both variants of the induced subgraph isomorphism problem. Under some assumption from parameterized complexity theory, these problems are solvable in polynomial time if and only if contains no arbitrarily large graphs. All classifications are given by means of parameterized complexity. The results are presented for arbitrary structures of bounded arity which implies, for example, analogous results for directed graphs. Furthermore, we show that no such dichotomy is possible in the sense of classical complexity. That is, if there are classes such that the induced subgraph isomorphism problem on is neither in nor -complete. This argument may be of independent interest, because it is applicable to various parameterized problems.

DECIDABILITY RESULTS FOR THE BOUNDEDNESS PROBLEM

We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones.

Decidability of S1S and S2S

The purpose of this chapter is to prove the decidability of monadic second-order logic over infinite words and infinite binary trees, denoted S1S and S2S. It is organized as follows.

Boundedness of Monadic Second-Order Formulae over Finite Words

We prove that the boundedness problem for monadic second-order logic over the class of all finite words is decidable.

Tree-width for first order formulae

We introduce tree-width for first order formulae \phi, fotw(\phi). We show that computing fotw is fixed-parameter tractable with parameter fotw. Moreover, we show that on classes of formulae of bounded fotw, model checking is fixed parameter tractable, with parameter the length of the formula. This is done by translating a formula \phi\ with fotw(\phi)<k into a formula of the k-variable fragment L^k of first order logic. For fixed k, the question whether a given first order formula is equivalent to an L^k formula is undecidable. In contrast, the classes of first order formulae with bounded fotw are fragments of first order logic for which the equivalence is decidable. Our notion of tree-width generalises tree-width of conjunctive queries to arbitrary formulae of first order logic by taking into account the quantifier interaction in a formula. Moreover, it is more powerful than the notion of elimination-width of quantified constraint formulae, defined by Chen and Dalmau (CSL 2005): for quantified constraint formulae, both bounded elimination-width and bounded fotw allow for model checking in polynomial time. We prove that fotw of a quantified constraint formula \phi\ is bounded by the elimination-width of \phi, and we exhibit a class of quantified constraint formulae with bounded fotw, that has unbounded elimination-width. A similar comparison holds for strict tree-width of non-recursive stratified datalog as defined by Flum, Frick, and Grohe (JACM 49, 2002). Finally, we show that fotw has a characterization in terms of a cops and robbers game without monotonicity cost.

Decidable relationships between consistency notions for constraint satisfaction problems

We define an abstract pebble game that provides game interpretations for essentially all known consistency algorithms for constraint satisfaction problems including arc-consistency, (j, k)-consistency, k-consistency, k-minimality, and refinements of arc-consistency such as peek arc-consistency and singleton arc-consistency. Our main result is that for any two instances of the abstract pebble game where the first satisfies the additional condition of being stacked, there exists an algorithm to decide whether consistency with respect to the first implies consistency with respect to the second. In particular, there is a decidable criterion to tell whether singleton arc-consistency with respect to a given constraint language implies k-consistency with respect to the same constraint language, for any fixed k. We also offer a new decidable criterion to tell whether arc-consistency implies satisfiability which pulls from methods in Ramsey theory and looks more amenable to generalization.

Bounded Fixed-Parameter Tractability: The Case 2poly( k)

We introduce a notion of fixed-parameter tractability permitting a parameter dependence of only 2poly(k). We delve into the corresponding intractability theory. In this course we define the PW-hierarchy, provide some complete problems, and characterize each of its classes in terms of model-checking problems, in terms of propositional satisfiability problems, in terms of logical definitions, and by machine models.

Bounded fixed-parameter tractability and reducibility

Abstract We study a refined framework of parameterized complexity theory where the parameter dependence of fixed-parameter tractable algorithms is not arbitrary, but restricted by a function in some family ℱ . For every family ℱ of functions, this yields a notion of ℱ -fixed-parameter tractability. If ℱ is the class of all polynomially bounded functions, then ℱ -fixed-parameter tractability coincides with polynomial time decidability and if ℱ is the class of all computable functions, ℱ -fixed-parameter tractability coincides with the standard notion of fixed-parameter tractability. There are interesting choices of ℱ between these two extremes, for example the class of all singly exponential functions. In this article, we study the general theory of ℱ -fixed-parameter tractability. We introduce a generic notion of reduction and two classes ℱ -W[P] and ℱ -XP , which may be viewed as analogues of NP and EXPTIME, respectively, in the world of ℱ -fixed-parameter tractability.