347 research outputs found
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.Comment: This paper extends and updates a paper that has been presented on the
workshop TAASP'16 (arXiv:1612.07601). We provide a higher detail level, full
proofs and more example
Tractable Optimization Problems through Hypergraph-Based Structural Restrictions
Several variants of the Constraint Satisfaction Problem have been proposed
and investigated in the literature for modelling those scenarios where
solutions are associated with some given costs. Within these frameworks
computing an optimal solution is an NP-hard problem in general; yet, when
restricted over classes of instances whose constraint interactions can be
modelled via (nearly-)acyclic graphs, this problem is known to be solvable in
polynomial time. In this paper, larger classes of tractable instances are
singled out, by discussing solution approaches based on exploiting hypergraph
acyclicity and, more generally, structural decomposition methods, such as
(hyper)tree decompositions
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
Counting Complexity for Reasoning in Abstract Argumentation
In this paper, we consider counting and projected model counting of
extensions in abstract argumentation for various semantics. When asking for
projected counts we are interested in counting the number of extensions of a
given argumentation framework while multiple extensions that are identical when
restricted to the projected arguments count as only one projected extension. We
establish classical complexity results and parameterized complexity results
when the problems are parameterized by treewidth of the undirected
argumentation graph. To obtain upper bounds for counting projected extensions,
we introduce novel algorithms that exploit small treewidth of the undirected
argumentation graph of the input instance by dynamic programming (DP). Our
algorithms run in time double or triple exponential in the treewidth depending
on the considered semantics. Finally, we take the exponential time hypothesis
(ETH) into account and establish lower bounds of bounded treewidth algorithms
for counting extensions and projected extension.Comment: Extended version of a paper published at AAAI-1
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
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