Stephan Kreutzer

DAG-Width and parity games

Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm development. Tree-width is characterised by a game known as the cops-and-robber game where a number of cops chase a robber on the graph. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure with an associated notion of graph decomposition that can be seen to describe how close a directed graph is to a directed acyclic graph (DAG). This promises to be useful in developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded DAG-width. We also consider the relationship between DAG-width and other measures such as entanglement and directed tree-width. One consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomial-time computable on graphs of bounded DAG-width.

Locally Excluding a Minor

We introduce the concept of locally excluded minors. Graph classes locally excluding a minor are a common generalisation of the concept of excluded minor classes and of graph classes with bounded local tree-width. We show that first-order model-checking is fixed-parameter tractable on any class of graphs locally excluding a minor. This strictly generalises analogous results by Flum and Grohe on excluded minor classes and Frick and Grohe on classes with bounded local tree-width. As an important consequence of the proof we obtain fixed-parameter algorithms for problems such as dominating or independent set on graph classes excluding a minor, where now the parameter is the size of the dominating set and the excluded minor. We also study graph classes with excluded minors, where the minor may grow slowly with the size of the graphs and show that again, first-order model-checking is fixed-parameter tractable on any such class of graphs.

Digraph measures: Kelly decompositions, games, and orderings

We consider various well-known, equivalent complexity measures for graphs such as elimination orderings, κ-trees and cops and robber games and study their natural translations to digraphs. We show that on digraphs all these measures are also equivalent and induce a natural connectivity measure. We introduce a decomposition for digraphs and an associated width, Kelly-width, which is equivalent to the aforementioned measure. We demonstrate its usefulness by exhibiting a number of potential applications including polynomial-time algorithms for NP-complete problems on graphs of bounded Kelly-width, and complexity analysis of asymmetric matrix factorization. Finally, we compare the new width to other known decompositions of digraphs.

The dag-width of directed graphs

Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm design. Tree-width can be characterised by a graph searching game where a number of cops attempt to capture a robber. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure, called dag-width, that can be seen to describe how close a directed graph is to a directed acyclic graph (dag). We also provide an associated decomposition and show how it is useful for developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded dag-width. We also consider the relationship between dag-width and other connectivity measures such as directed tree-width and path-width. A consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomial-time computable on graphs of bounded dag-width.

Reachability in Succinct and Parametric One-Counter Automata

One-counter automata are a fundamental and widely-studied class of infinite-state systems. In this paper we consider one-counter automata with counter updates encoded in binary--which we refer to as the succinct encoding. It is easily seen that the reachability problem for this class of machines is in PSpace and is NP -hard. One of the main results of this paper is to show that this problem is in fact in NP , and is thus NP -complete. We also consider parametric one-counter automata, in which counter updates be integer-valued parameters. The reachability problem asks whether there are values for the parameters such that a final state can be reached from an initial state. Our second main result shows decidability of the reachability problem for parametric one-counter automata by reduction to existential Presburger arithmetic with divisibility.

Algorithmic meta-theorems

Algorithmic meta-theorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class Open image in new window of problems can be solved efficiently on every graph satisfying a certain property Open image in new window ”. A particularly well known example of a meta-theorem is Courcelle’s theorem that every decision problem definable in monadic second-order logic (MSO) can be decided in linear time on any class of graphs of bounded tree-width [1].

Inflationary fixed points in modal logic

We consider an extension of modal logic with an operator for constructing inflationary fixed points, just as the modal μ-calculus extends basic modal logic with an operator for least fixed points. Least and inflationary fixed-point operators have been studied and compared in other contexts, particularly in finite model theory, where it is known that the logics IFP and LFP that result from adding such fixed-point operators to first-order logic have equal expressive power. As we show, the situation in modal logic is quite different, as the modal iteration calculus (MIC), we introduce has much greater expressive power than the μ-calculus. Greater expressive power comes at a cost: the calculus is algorithmically much less manageable.

Model theory makes formulas large

Gaifmans locality theorem states that every first-order sentence is equivalent to a local sentence. We show that there is no elementary bound on the length of the local sentence in terms of the original. The classical Łoś-Tarski theorem states that every first-order sentence preserved under extensions is equivalent to an existential sentence. We show that there is no elementary bound on the length of the existential sentence in terms of the original. Recently, variants of the Łoś-Tarski theorem have been proved for certain classes of finite structures, among them the class of finite acyclic structures and more generally classes of structures of bounded tree width. Our lower bound also applies to these variants. We further prove that a version of the Feferman-Vaught theorem based on a restriction by formula length necessarily entails a non-elementary blow-up in formula size. All these results are based on a similar technique of encoding large numbers by trees of small height in such a way that small formulas can speak about these numbers. Notably, our lower bounds do not apply to restrictions of the results to structures of bounded degree. For such structures, we obtain elementary upper bounds in all cases. However, even there we can prove at least doubly exponential lower bounds.

Approximation Schemes for First-Order Definable Optimisation Problems

Let phi(X) be a first-order formula in the language of graphs that has a free set variable X, and assume that X only occurs positively in phi(X). Then a natural minimisation problem associated with phi(X) is to find, in a given graph G, a vertex set S of minimum size such that G satisfies phi(S). Similarly, if X only occurs negatively in phi(X), then phi(X) defines a maximisation problem. Many well-known optimisation problems are first-order definable in this sense, for example, minimum dominating set or maximum independent set. We prove that for each class Gscr of graphs with excluded minors, in particular for each class of planar graphs, the restriction of a first-order definable optimisation problem to the class Gscr has a polynomial time approximation scheme. A crucial building block of the proof of this approximability result is a version of Gaifmans locality theorem for formulas positive in a set variable. This result may be of independent interest

Domination Problems in Nowhere-Dense Classes

We investigate the parameterized complexity of generalisations and variations of the dominating set problem on classes of graphs that are nowhere dense. In particular, we show that the distance-