Stefan Rümmele

Counting and enumeration problems with bounded treewidth

By Courcelles Theorem we know that any property of finite structures definable in monadic second-order logic (MSO) becomes tractable over structures with bounded treewidth. This result was extended to counting problems by Arnborg et al. and to enumeration problems by Flum et al. Despite the undisputed importance of these results for proving fixed-parameter tractability, they do not directly yield implementable algorithms. Recently, Gottlob et al. presented a new approach using monadic datalog to close the gap between theoretical tractability and practical computability for MSO-definable decision problems. In the current work we show how counting and enumeration problems can be tackled by an appropriate extension of the datalog approach.

Tractable answer-set programming with weight constraints: bounded treewidth is not enough

Cardinality constraints or, more generally, weight constraints are well recognized as an important extension of answer-set programming. Clearly, all common algorithmic tasks related to programs with cardinality or weight constraints - like checking the consistency of a program - are intractable. Many intractable problems in the area of knowledge representation and reasoning have been shown to become linear time tractable if the treewidth of the programs or formulas under consideration is bounded by some constant. The goal of this paper is to apply the notion of treewidth to programs with cardinality or weight constraints and to identify tractable fragments. It will turn out that the straightforward application of treewidth to such class of programs does not suffice to obtain tractability. However, by imposing further restrictions, tractability can be achieved.

Fast Counting with Bounded Treewidth

Many intractable problems have been shown to become tractable if the treewidth of the underlying structure is bounded by a constant. An important tool for deriving such results is Courcelles Theorem, which states that all properties defined by Monadic-Second Order (MSO) sentences are fixed-parameter tractable with respect to the treewidth. Arnborg et al. extended this result to counting problems defined via MSO properties. However, the MSO description of a problem is of course not an algorithm. Consequently, proving the fixed-parameter tractability of some problem via Courcelles Theorem can be considered as the starting point rather than the endpoint of the search for an efficient algorithm. Gottlob et al. have recently presented a new approach via monadic datalog to actually devise efficient algorithms for decision problems whose tractability follows from Courcelles Theorem. In this paper, we extend this approach and apply it to some fundamental counting problems in logic an artificial intelligence.

Merging in the Horn Fragment

Belief merging is a central operation within the field of belief change and addresses the problem of combining multiple, possibly mutually inconsistent knowledge bases into a single, consistent one. A current research trend in belief change is concerned with representation theorems tailored to fragments of logic, in particular Horn logic. Hereby, the goal is to guarantee that the result of the change operations stays within the fragment under consideration. While several such results have been obtained for Horn revision and Horn contraction, merging of Horn theories has been neglected so far. In this article, we provide a novel representation theorem for Horn merging by strengthening the standard merging postulates. Moreover, we present concrete Horn merging operators satisfying all postulates.

Multicut algorithms via tree decompositions

Various forms of multicut problems are of great importance in the area of network design. In general, these problems are intractable. However, several parameters have been identified which lead to fixed-parameter tractability (FPT). Recently, Gottlob and Lee have proposed the treewidth of the structure representing the graph and the set of pairs of terminal vertices as one such parameter. In this work, we show how this theoretical FPT result can be turned into efficient algorithms for optimization, counting, and enumeration problems in this area.

Evaluating Tree-Decomposition Based Algorithms for Answer Set Programming

A promising approach to tackle intractable problems is given by a combination of decomposition methods with dynamic algorithms. One such decomposition concept is tree decomposition. However, several heuristics for obtaining a tree decomposition exist and, moreover, also the subsequent dynamic algorithm can be laid out differently. In this paper, we provide an experimental evaluation of this combined approach when applied to reasoning problems in propositional answer set programming. More specifically, we analyze the performance of three different heuristics and two different dynamic algorithms, an existing standard version and a recently proposed algorithm based on a more involved data structure, but which provides better theoretical runtime. The results suggest that a suitable combination of the tree decomposition heuristics and the dynamic algorithm has to be chosen carefully. In particular, we observed that the performance of the dynamic algorithm highly depends on certain features (besides treewidth) of the provided tree decomposition. Based on this observation we apply supervised machine learning techniques to automatically select the dynamic algorithm depending on the features of the input tree decomposition.

Belief Merging within Fragments of Propositional Logic

Recently, belief change within the framework of fragments of propositional logic has gained increasing attention. Previous research focused on belief contraction and belief revision on the Horn fragment. However, the problem of belief merging within fragments of propositional logic has been mostly neglected so far. We present a general approach to defining new merging operators derived from existing ones such that the result of merging remains in the fragment under consideration. Our approach is not limited to the case of Horn fragment; it is applicable to any fragment of propositional logic characterized by a closure property on the sets of models of its formulæ. We study the logical properties of the proposed operators regarding satisfaction of merging postulates, considering, in particular, distance-based merging operators for Horn and Krom fragments.

A dynamic-programming based ASP-solver

We present a novel system for propositional Answer-Set Programming (ASP). This system, called dynASP, is based on dynamic programming and thus significantly differs from standard ASP-solvers which implement techniques stemming from SAT or CSP.

Belief Revision with Bounded Treewidth

Problems arising from the revision of propositional knowledge bases have been intensively studied for two decades. Many different approaches to revision have thus been suggested, with the ones by Dalal or Satoh being two of the most fundamental ones. As is well known, most computational tasks in this area are intractable. Therefore, in practical applications, one requires sufficient conditions under which revision problems become efficiently solvable. In this paper, we identify such tractable fragments for the reasoning and the enumeration problem exploiting the notion of treewidth. More specifically, we present new algorithms based on dynamic programming for these problems in Dalals setting and a tractability proof using Courcelles Theorem for Satohs approach.

Multicut on Graphs of Bounded Clique-Width

Several variants of Multicut problems arise in applications like circuit and network design. In general, these problems are NP-complete. The goal of our work is to investigate the potential of clique-width for identifying tractable fragments of Multicut. We show for several parameterizations involving clique-width whether they lead to tractability or not. Since bounded tree-width implies bounded clique-width, our tractability results extend previous results via tree-width, in particular to dense graphs.