16 research outputs found
Backdoors to Normality for Disjunctive Logic Programs
Over the last two decades, propositional satisfiability (SAT) has become one
of the most successful and widely applied techniques for the solution of
NP-complete problems. The aim of this paper is to investigate theoretically how
Sat can be utilized for the efficient solution of problems that are harder than
NP or co-NP. In particular, we consider the fundamental reasoning problems in
propositional disjunctive answer set programming (ASP), Brave Reasoning and
Skeptical Reasoning, which ask whether a given atom is contained in at least
one or in all answer sets, respectively. Both problems are located at the
second level of the Polynomial Hierarchy and thus assumed to be harder than NP
or co-NP. One cannot transform these two reasoning problems into SAT in
polynomial time, unless the Polynomial Hierarchy collapses. We show that
certain structural aspects of disjunctive logic programs can be utilized to
break through this complexity barrier, using new techniques from Parameterized
Complexity. In particular, we exhibit transformations from Brave and Skeptical
Reasoning to SAT that run in time O(2^k n^2) where k is a structural parameter
of the instance and n the input size. In other words, the reduction is
fixed-parameter tractable for parameter k. As the parameter k we take the size
of a smallest backdoor with respect to the class of normal (i.e.,
disjunction-free) programs. Such a backdoor is a set of atoms that when deleted
makes the program normal. In consequence, the combinatorial explosion, which is
expected when transforming a problem from the second level of the Polynomial
Hierarchy to the first level, can now be confined to the parameter k, while the
running time of the reduction is polynomial in the input size n, where the
order of the polynomial is independent of k.Comment: A short version will appear in the Proceedings of the Proceedings of
the 27th AAAI Conference on Artificial Intelligence (AAAI'13). A preliminary
version of the paper was presented on the workshop Answer Set Programming and
Other Computing Paradigms (ASPOCP 2012), 5th International Workshop,
September 4, 2012, Budapest, Hungar
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
A Paraconsistent ASP-like Language with Tractable Model Generation
Answer Set Programming (ASP) is nowadays a dominant rule-based knowledge
representation tool. Though existing ASP variants enjoy efficient
implementations, generating an answer set remains intractable. The goal of this
research is to define a new \asp-like rule language, 4SP, with tractable model
generation. The language combines ideas of ASP and a paraconsistent rule
language 4QL. Though 4SP shares the syntax of \asp and for each program all its
answer sets are among 4SP models, the new language differs from ASP in its
logical foundations, the intended methodology of its use and complexity of
computing models.
As we show in the paper, 4QL can be seen as a paraconsistent counterpart of
ASP programs stratified with respect to default negation. Although model
generation of well-supported models for 4QL programs is tractable, dropping
stratification makes both 4QL and ASP intractable. To retain tractability while
allowing non-stratified programs, in 4SP we introduce trial expressions
interlacing programs with hypotheses as to the truth values of default
negations. This allows us to develop a~model generation algorithm with
deterministic polynomial time complexity.
We also show relationships among 4SP, ASP and 4QL
Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning under structural
restrictions. All these problems involve two tasks: (i) identifying the
structure in the input as required by the restriction, and (ii) using the
identified structure to solve the reasoning task efficiently. We show that for
most of the considered problems, task (i) admits a polynomial-time
preprocessing to a problem kernel whose size is polynomial in a structural
problem parameter of the input, in contrast to task (ii) which does not admit
such a reduction to a problem kernel of polynomial size, subject to a
complexity theoretic assumption. As a notable exception we show that the
consistency problem for the AtMost-NValue constraint admits a polynomial kernel
consisting of a quadratic number of variables and domain values. Our results
provide a firm worst-case guarantees and theoretical boundaries for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541,
arXiv:1104.556
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
Lower Bounds for QBFs of Bounded Treewidth
The problem of deciding the validity (QSAT) of quantified Boolean formulas
(QBF) is a vivid research area in both theory and practice. In the field of
parameterized algorithmics, the well-studied graph measure treewidth turned out
to be a successful parameter. A well-known result by Chen in parameterized
complexity is that QSAT when parameterized by the treewidth of the primal graph
of the input formula together with the quantifier depth of the formula is
fixed-parameter tractable. More precisely, the runtime of such an algorithm is
polynomial in the formula size and exponential in the treewidth, where the
exponential function in the treewidth is a tower, whose height is the
quantifier depth. A natural question is whether one can significantly improve
these results and decrease the tower while assuming the Exponential Time
Hypothesis (ETH). In the last years, there has been a growing interest in the
quest of establishing lower bounds under ETH, showing mostly problem-specific
lower bounds up to the third level of the polynomial hierarchy. Still, an
important question is to settle this as general as possible and to cover the
whole polynomial hierarchy. In this work, we show lower bounds based on the ETH
for arbitrary QBFs parameterized by treewidth (and quantifier depth). More
formally, we establish lower bounds for QSAT and treewidth, namely, that under
ETH there cannot be an algorithm that solves QSAT of quantifier depth i in
runtime significantly better than i-fold exponential in the treewidth and
polynomial in the input size. In doing so, we provide a versatile reduction
technique to compress treewidth that encodes the essence of dynamic programming
on arbitrary tree decompositions. Further, we describe a general methodology
for a more fine-grained analysis of problems parameterized by treewidth that
are at higher levels of the polynomial hierarchy