Markus Hecher

Answer Set Solving with Bounded Treewidth Revisited

Parameterized algorithms are a way to solve hard problems more efficiently, given that a specific parameter of the input is small. In this paper, we apply this idea to the field of answer set programming (ASP). To this end, we propose two kinds of graph representations of programs to exploit their treewidth as a parameter. Treewidth roughly measures to which extent the internal structure of a program resembles a tree. Our main contribution is the design of parameterized dynamic programming algorithms, which run in linear time if the treewidth and weights of the given program are bounded. Compared to previous work, our algorithms handle the full syntax of ASP. Finally, we report on an empirical evaluation that shows good runtime behaviour for benchmark instances of low treewidth, especially for counting answer sets.

The D-FLAT System for Dynamic Programming on Tree Decompositions

Complex reasoning problems over large amounts of data pose a great challenge for computer science. To overcome the obstacle of high computational complexity, exploiting structure by means of tree decompositions has proved to be effective in many cases. However, the implementation of suitable efficient algorithms is often tedious. D-FLAT is a software system that combines the logic programming language Answer Set Programming with problem solving on tree decompositions and can serve as a rapid prototyping tool for such algorithms. Since we initially proposed D-FLAT, we have made major changes to the system, improving its range of applicability and its usability. In this paper, we present the system resulting from these efforts.

D-FLAT^2: Subset Minimization in Dynamic Programming on Tree Decompositions Made Easy

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 D-FLATˆ2, a system that allows one to obtain DP algorithms (specified in ASP) from simpler principles, where the DP formalization of subset-minimization is performed automatically. We illustrate the method at work by providing several DP algorithms – given in form of ASP programs – 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.

DynASP2.5: Dynamic Programming on Tree Decompositions in Action

A vibrant theoretical research area are efficient exact parameterized algorithms. Very recent solving competitions such as the PACE challenge show that there is also increasing practical interest in the parameterized algorithms community. An important research question is whether dedicated parameterized exact algorithms exhibit certain practical relevance and one can even beat well-established problem solvers. We consider the logic-based declarative modeling language and problem solving framework Answer Set Programming (ASP). State-of-the-art ASP solvers rely considerably on Sat-based algorithms. An ASP solver (DynASP2), which is based on a classical dynamic programming on tree decompositions, has been published very recently. Unfortunately, DynASP2 can outperform modern ASP solvers on programs of small treewidth only if the question of interest is to count the number of solutions. In this paper, we describe underlying concepts of our new implementation (DynASP2.5) that shows competitive behavior to state-of-the-art ASP solvers even for finding just one solution when solving problems as the Steiner tree problem that have been modeled in ASP on graphs with low treewidth. Our implementation is based on a novel approach that we call multi-pass dynamic programming (M-DPSINC).

Exploiting Treewidth for Projected Model Counting and Its Limits.

In this paper, we introduce a novel algorithm to solve projected model counting (PMC). PMC asks to count solutions of a Boolean formula with respect to a given set of projected variables, where multiple solutions that are identical when restricted to the projected variables count as only one solution. Our algorithm exploits small treewidth of the primal graph of the input instance. It runs in time \({\mathcal O}(2^{2^{k+4}} n^2)\) where k is the treewidth and n is the input size of the instance. In other words, we obtain that the problem PMC is fixed-parameter tractable when parameterized by treewidth. Further, we take the exponential time hypothesis (ETH) into consideration and establish lower bounds of bounded treewidth algorithms for PMC, yielding asymptotically tight runtime bounds of our algorithm.

An SMT Approach to Fractional Hypertree Width

Bounded fractional hypertree width ( Open image in new window ) is the most general known structural property that guarantees polynomial-time solvability of the constraint satisfaction problem. Bounded Open image in new window generalizes other structural properties like bounded induced width and bounded hypertree width.

Default Logic and Bounded Treewidth

In this paper, we study Reiters propositional default logic when the treewidth of a certain graph representation (semi-primal graph) of the input theory is bounded. We establish a dynamic programming algorithm on tree decompositions that decides whether a theory has a consistent stable extension (Ext). Our algorithm can even be used to enumerate all generating defaults (ExtEnum) that lead to stable extensions. We show that our algorithm decides Ext in linear time in the input theory and triple exponential time in the treewidth (so-called fixed-parameter linear algorithm). Further, our algorithm solves ExtEnum with a pre-computation step that is linear in the input theory and triple exponential in the treewidth followed by a linear delay to output solutions.

Dynamic Programming on Tree Decompositions with D-FLAT

Many hard problems can be solved efficiently by dynamic programming algorithms that work on tree decompositions. In this paper, we present the D-FLAT system for rapid prototyping of such algorithms. Users can specify the algorithm for their problem using Answer Set Programming. We illustrate the framework by an example and briefly discuss its main features.