20 research outputs found
Treewidth in Non-Ground Answer Set Solving and Alliance Problems in Graphs
To solve hard problems efficiently via answer set programming (ASP), a promising approach is to take advantage of the fact that real-world instances of many hard problems exhibit small treewidth. Algorithms that exploit this have already been proposed -- however, they suffer from an enormous overhead. In the thesis, we present improvements in the algorithmic methodology for leveraging bounded treewidth that are especially targeted toward problems involving subset minimization. This can be useful for many problems at the second level of the polynomial hierarchy like solving disjunctive ground ASP. Moreover, we define classes of non-ground ASP programs such that grounding such a program together with input facts does not lead to an excessive increase in treewidth of the resulting ground program when compared to the treewidth of the input. This allows ASP users to take advantage of the fact that state-of-the-art ASP solvers perform better on ground programs of small treewidth. Finally, we resolve several open questions on the complexity of alliance problems in graphs. In particular, we settle the long-standing open questions of the complexity of the Secure Set problem and whether the Defensive Alliance problem is fixed-parameter tractable when parameterized by treewidth
Treewidth-aware Reductions of Normal ASP to SAT -- Is Normal ASP Harder than SAT after All?
Answer Set Programming (ASP) is a paradigm for modeling and solving problems
for knowledge representation and reasoning. There are plenty of results
dedicated to studying the hardness of (fragments of) ASP. So far, these studies
resulted in characterizations in terms of computational complexity as well as
in fine-grained insights presented in form of dichotomy-style results, lower
bounds when translating to other formalisms like propositional satisfiability
(SAT), and even detailed parameterized complexity landscapes. A generic
parameter in parameterized complexity originating from graph theory is the
so-called treewidth, which in a sense captures structural density of a program.
Recently, there was an increase in the number of treewidth-based solvers
related to SAT. While there are translations from (normal) ASP to SAT, no
reduction that preserves treewidth or at least keeps track of the treewidth
increase is known. In this paper we propose a novel reduction from normal ASP
to SAT that is aware of the treewidth, and guarantees that a slight increase of
treewidth is indeed sufficient. Further, we show a new result establishing
that, when considering treewidth, already the fragment of normal ASP is
slightly harder than SAT (under reasonable assumptions in computational
complexity). This also confirms that our reduction probably cannot be
significantly improved and that the slight increase of treewidth is
unavoidable. Finally, we present an empirical study of our novel reduction from
normal ASP to SAT, where we compare treewidth upper bounds that are obtained
via known decomposition heuristics. Overall, our reduction works better with
these heuristics than existing translations
DynASP2.5: Dynamic Programming on Tree Decompositions in Action
Efficient, exact parameterized algorithms are a vibrant theoretical research area. 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 such algorithms can be built to efficiently solve specific problems in practice, that is, to be competitive with established solving systems. In this paper, we consider Answer Set Programming (ASP), a logic-based declarative modeling and problem solving framework. State-of-the-art ASP solvers generally rely on SAT-based algorithms. In addition, DynASP2, an ASP solver that is based on a classical dynamic programming on tree decompositions, has recently been introduced. DynASP2 outperforms modern ASP solvers when the goal is to count the number of solutions of programs that have small treewidth. However, for quickly finding one solutions, DynASP2 proved uncompetitive. In this paper, we present a new algorithm and implementation, called DynASP2.5, that shows competitive behavior compared to state-of-the-art ASP solvers on problems like Steiner tree for low-treewidth graphs, even when the
task is to find just one solution. Our implementation is based on a novel approach that we call multi-pass dynamic programming
Treewidth-Aware Complexity in ASP: Not all Positive Cycles are Equally Hard
It is well-know that deciding consistency for normal answer set programs
(ASP) is NP-complete, thus, as hard as the satisfaction problem for classical
propositional logic (SAT). The best algorithms to solve these problems take
exponential time in the worst case. The exponential time hypothesis (ETH)
implies that this result is tight for SAT, that is, SAT cannot be solved in
subexponential time. This immediately establishes that the result is also tight
for the consistency problem for ASP. However, accounting for the treewidth of
the problem, the consistency problem for ASP is slightly harder than SAT: while
SAT can be solved by an algorithm that runs in exponential time in the
treewidth k, it was recently shown that ASP requires exponential time in k
\cdot log(k). This extra cost is due checking that there are no self-supported
true atoms due to positive cycles in the program. In this paper, we refine the
above result and show that the consistency problem for ASP can be solved in
exponential time in k \cdot log({\lambda}) where {\lambda} is the minimum
between the treewidth and the size of the largest strongly-connected component
in the positive dependency graph of the program. We provide a dynamic
programming algorithm that solves the problem and a treewidth-aware reduction
from ASP to SAT that adhere to the above limit
Current and Future Challenges in Knowledge Representation and Reasoning
Knowledge Representation and Reasoning is a central, longstanding, and active
area of Artificial Intelligence. Over the years it has evolved significantly;
more recently it has been challenged and complemented by research in areas such
as machine learning and reasoning under uncertainty. In July 2022 a Dagstuhl
Perspectives workshop was held on Knowledge Representation and Reasoning. The
goal of the workshop was to describe the state of the art in the field,
including its relation with other areas, its shortcomings and strengths,
together with recommendations for future progress. We developed this manifesto
based on the presentations, panels, working groups, and discussions that took
place at the Dagstuhl Workshop. It is a declaration of our views on Knowledge
Representation: its origins, goals, milestones, and current foci; its relation
to other disciplines, especially to Artificial Intelligence; and on its
challenges, along with key priorities for the next decade
Description Logic for Scene Understanding at the Example of Urban Road Intersections
Understanding a natural scene on the basis of external sensors is a task yet to be solved by computer algorithms. The present thesis investigates the suitability of a particular family of explicit, formal representation and reasoning formalisms for this task, which are subsumed under the term Description Logic