Martin Grohe

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.

Path queries on compressed XML

Central to any XML query language is a path language such as XPath which operates on the tree structure of the XML document. We demonstrate in this paper that the tree structure can be effectively compressed and manipulated using techniques derived from symbolic model checking. Specifically, we show first that succinct representations of document tree structures based on sharing subtrees are highly effective. Second, we show that compressed structures can be queried directly and efficiently through a process of manipulating selections of nodes and partial decompression. We study both the theoretical and experimental properties of this technique and provide algorithms for querying our compressed instances using node-selecting path query languages such as XPath. We believe the ability to store and manipulate large portions of the structure of very large XML documents in main memory is crucial to the development of efficient, scalable native XML databases and query engines.

Deciding first-order properties of locally tree-decomposable structures

We introduce the concept of a class of graphs, or more generally, relational structures, being locally tree-decomposable. There are numerous examples of locally tree-decomposable classes, among them the class of planar graphs and all classes of bounded valence or of bounded tree-width. We also consider a slightly more general concept of a class of structures having bounded local tree-width.We show that for each property φ of structures that is definable in first-order logic and for each locally tree-decomposable class C of structures, there is a linear time algorithm deciding whether a given structure A ∈ C has property φ. For classes C of bounded local tree-width, we show that for every k ≥ 1 there is an algorithm solving the same problem in time O(n1+(1/k)) (where n is the cardinality of the input structure).

Constraint solving via fractional edge covers

Many important combinatorial problems can be modelled as constraint satisfaction problems, hence identifying polynomial-time solvable classes of constraint satisfaction problems received a lot of attention. In this paper, we are interested in structural properties that can make the problem tractable. So far, the largest structural class that is known to be polynomial-time solvable is the class of bounded hypertree width instances introduced by Gottlob et al. [20]. Here we identify a new class of polynomial-time solvable instances: those having bounded fractional edge cover number.Combining hypertree width and fractional edge cover number, we then introduce the notion of fractional hypertree width. We prove that constraint satisfaction problems with bounded fractional hypertree width can be solved in polynomial time (provided that a the tree decomposition is given in the input). We also prove that certain parameterized constraint satisfaction, homomorphism, and embedding problems are fixed-parameter tractable on instances having bounded fractional hypertree width.

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

The complexity of partition functions

[t]

When is the evaluation of conjunctive queries tractable

, for

Locally Excluding a Minor

t\ge 1

Query evaluation on compressed trees

, 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

Finding topological subgraphs is fixed-parameter tractable

[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