In this paper, we aim to study how the performance of modern answer set programming (ASP) solvers is influenced by the treewidth of the input program and to investigate the consequences of this relationship. We first perform an experimental evaluation that shows that the solving performance is heavily influenced by treewidth, given ground input programs that are otherwise uniform, both in size and construction. This observation leads to an important question for ASP, namely, how to design encodings such that the treewidth of the resulting ground program remains small. To this end, we study two classes of disjunctive programs, namely guarded and connection-guarded programs. In order to investigate these classes, we formalize the grounding process using MSO transductions. Our main results show that both classes guarantee that the treewidth of the program after grounding only depends on the treewidth (and the maximum degree, in case of connection-guarded programs) of the input instance. In terms of parameterized complexity, our findings yield corresponding FPT results for answer-set existence for bounded treewidth (and also degree, for connection-guarded programs) of the input instance. We further show that bounding treewidth alone leads to NP-hardness in the data complexity for connection-guarded programs, which indicates that the two classes are fundamentally different. Finally, we show that for both classes, the data complexity remains as hard as in the general case of ASP.

Abstract The notion of secure sets is a rather new concept in the area of graph theory. Applied to social network analysis, the goal is to identify groups of entities that can repel any attack or influence from the outside. In this article, we tackle this problem by utilizing Answer Set Programming (ASP). It is known that verifying whether a set is secure in a graph is already co-NP-hard. Therefore, the problem of enumerating all secure sets is challenging for ASP and its systems. In particular, encodings for this problem seem to require disjunction and also recursive aggregates. Here, we provide such encodings and analyse their performance using the Clingo system. Furthermore, we study several problem variants, including multiple secure or insecure sets, and weighted graphs.

Model counting is the archetypical #P-complete problem consisting of determining the number of satisfying truth assignments of a given propositional formula. In this short paper, we empirically investigate the potential of employing graph centrality measures as a basis of search heuristics in the context of exact model counting. In particular, we integrate centrality-based heuristics into the search-based exact model counter sharpSAT. Our experiments show that employing centrality information significantly improves the empirical performance of sharpSAT, and also allows for simplifying the search heuristics compared to the current default heuristics of the model counter. In particular, we show that the VSIDS heuristic, which is an integral search heuristic employed in essentially all state-of-the-art conflict-driven clause learning Boolean satisfiability solvers, appears to be of very limited use in the context of model counting.

In this paper, we aim to study how the performance of modern answer set programming (ASP) solvers is influenced by the treewidth of the input program and to investigate the consequences of this relationship. We first perform an experimental evaluation that shows that the solving performance is heavily influenced by the treewidth, given ground input programs that are otherwise uniform, both in size and construction. This observation leads to an important question for ASP, namely, how to design encodings such that the treewidth of the resulting ground program remains small. To this end, we define the class of connection-guarded programs, which guarantees that the treewidth of the program after grounding only depends on the treewidth (and the degree) of the input instance. In order to obtain this result, we formalize the grounding process using MSO transductions.

Many computationally hard problems become tractable if the graph structure underlying the problem instance exhibits small treewidth. A recent approach to put this idea into practice is based on a declarative interface to Answer Set Programming that allows us to specify dynamic programming over tree decompositions in this language, delegating the computation to dedicated solvers. In this article, we prove that this method can be applied to any problem whose fixed-parameter tractability follows from Courcelle's Theorem.

Many computational problems in the area of abstract argumentation are intractable. For some semantics like preferred and semi-stable, important decision problems can even be hard for classes of the second level of the polynomial hierarchy. One approach to deal with this inherent difficulty is to exploit structure of argumentation frameworks. In particular, algorithms that run in linear time for argumentation frameworks of bounded treewidth have been proposed for several semantics. In this paper, we contribute to this line of research and propose a novel algorithm for the semi-stable semantics. We also present an implementation of the algorithm and report on some experimental results.

Disjunctive Answer Set Programming (ASP) is a powerful declarative programming paradigm whose main decision problems are located on the second level of the polynomial hierarchy. Identifying tractable fragments and developing efficient algorithms for such fragments are thus important objectives in order to complement the sophisticated ASP systems available to date. Hard problems can become tractable if some problem parameter is bounded by a fixed constant; such problems are then called fixed-parameter tractable (FPT). While several FPT results for ASP exist, parameters that relate to directed or signed graphs representing the program at hand have been neglected so far. In this paper, we first give some negative observations showing that directed width measures on the dependency graph of a program do not lead to FPT results. We then consider the graph parameter of signed clique-width and present a novel dynamic programming algorithm that is FPT w.r.t. this parameter. Clique-width is more general than the well-known treewidth, and, to the best of our knowledge, ours is the first FPT algorithm for bounded clique-width for reasoning problems beyond SAT.

Many problems from the area of AI have been shown tractable for bounded treewidth. In order to put such results into practice, quite involved dynamic programming (DP) algorithms on tree decompositions have to be designed and implemented. These algorithms typically show recurring patterns that call for tasks like subset minimization. In this paper we present a novel approach to obtain such DP algorithms from simpler principles, where the DP formalization of subset minimization is performed automatically. We first give a theoretical account of our novel method, and then present D-FLAT^2, a system that allows one to specify the core DP algorithm via answer set programming (ASP). We illustrate the approach at work by providing several DP algorithms that are more space-efficient than existing solutions, while featuring improved readability, reuse and therefore maintainability of ASP code. Experiments show that our approach also yields a significant improvement in runtime performance.

A secure set S in a graph is defined as a set of vertices such that for any $$X\subseteq S$$ the majority of vertices in the neighborhood of X belongs to S. It is known that deciding whether a set S is secure in a graph is $$\text {co-}\hbox {NP}$$ -complete. However, it is still open how this result contributes to the actual complexity of deciding whether, for a given graph G and integer k, a non-empty secure set for G of size at most k exists. While membership in the class $$\Sigma ^{\mathrm{P}}_{2}$$ is rather easy to see for this existence problem, showing $$\Sigma ^{\mathrm{P}}_{2}$$ -hardness is quite involved. In this paper, we provide such a hardness result, hence classifying the secure set existence problem as $$\Sigma ^{\mathrm{P}}_{2}$$ -complete. We do so by first showing hardness for a variantof the problem, which we then reduce step-by-step to secure set existence. In total, we obtain eight new completeness results for different variants of the secure set existence problem.

Abstract In this work, we propose Answer-Set Programming (ASP) as a tool for rapid prototyping of dynamic programming algorithms based on tree decompositions. In fact, many such algorithms have been designed, but only a few of them found their way into implementation. The main obstacle is the lack of easy-to-use systems which (i) take care of building a tree decomposition and (ii) provide an interface for declarative specifications of dynamic programming algorithms. In this paper, we present D-FLAT, a novel tool that relieves the user of having to handle all the technical details concerned with parsing, tree decomposition, the handling of data structures, etc. Instead, it is only the dynamic programming algorithm itself which has to be specified in the ASP language. D-FLAT employs an ASP solver in order to compute the local solutions in the dynamic programming algorithm. In the paper, we give a few examples illustrating the use of D-FLAT and describe the main features of the system. Moreover, we report experiments which show that ASP-based D-FLAT encodings for some problems outperform monolithic ASP encodings on instances of small treewidth.