Jörg Flum

Query evaluation via tree-decompositions

A number of efficient methods for evaluating first-order and monadic-second order queries on finite relational structures are based on tree-decompositions of structures or queries. We systematically study these methods.In the first part of the article, we consider arbitrary formulas on tree-like structures. We generalize a theorem of Courcelle [1990] by showing that on structures of bounded tree-width a monadic second-order formula (with free first- and second-order variables) can be evaluated in time linear in the structure size plus the size of the output.In the second part, we study tree-like formulas on arbitrary structures. We generalize the notions of acyclicity and bounded tree-width from conjunctive queries to arbitrary first-order formulas in a straightforward way and analyze the complexity of evaluating formulas of these fragments. Moreover, we show that the acyclic and bounded tree-width fragments have the same expressive power as the well-known guarded fragment and the finite-variable fragments of first-order logic, respectively.

The Parameterized Complexity of Counting Problems

We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W

Logic and automata : history and perspectives

[t]

Query Evaluation via Tree-Decompositions

, for

Bounded fixed-parameter tractability and log2n nondeterministic bits

t\ge 1

On Parameterized Path and Chordless Path Problems

, that corresponds to Downey and Fellowss W-hierarchy [R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999] and we show that a few central W-completeness results for decision problems translate to \#W-completeness results for the corresponding counting problems. Counting complexity gets interesting with problems whose decision version is tractable, but whose counting version is hard. Our main result states that counting cycles and paths of length k in both directed and undirected graphs, parameterized by k, is #W

Machine-based methods in parameterized complexity theory

[1]-complete. This makes it highly unlikely that these problems are fixed-parameter tractable, even though their decision versions are fixed-parameter tractable. More explicitly, our result shows that most likely there is no

Total and Partial Well-Founded Datalog Coincide

f(k) \cdot n^c

Machine Characterizations of the Classes of the W-Hierarchy

-algorithm for counting cycles or paths of length k in a graph of size n for any computable function

On p-optimal proof systems and logics for PTIME

f: \mathbb{N} \to \mathbb{N}