Treewidth and Counting Projected Answer Sets

Fichte, Johannes Klaus; Hecher, Markus

doi:10.1007/978-3-030-20528-7_9

Cited by 19 publications

(33 citation statements)

References 33 publications

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“…In the following, we first use this theorem to establish a full methodology to obtain lower bound results for bounded treewidth and then provide a proof for the theorem in the next section. The result for k = 2 (cf., [44]) has already been applied as a strategy to show lower bound results for problems in artificial intelligence, as for example abstract argumentation, abduction, circumscription, and projected model counting, that are hard for the second level of the polynomial hierarchy when parameterized by treewidth [28,31,45]. With the generalization to an arbitrary quantifier depth in Theorem 13, one can obtain lower bounds for variants of these problems and even more general problems on the third level or higher levels of the polynomial hierarchy.…”

Section: Methodology For Lower Boundsmentioning

confidence: 99%

Lower Bounds for QBFs of Bounded Treewidth

Fichte¹,

Hecher²,

Pfandler³

2019

Preprint

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The problem of deciding the validity (QSAT) of quantified Boolean formulas (QBF) is a vivid research area with strong interest in practical as well as theoretical advances. This is witnessed by various solvers competing in the annual QBF evaluation as well as a notable progress in the theoretical understanding of the problem. One reason for the interest in QBF is that it is the prototypical problem for the polynomial hierarchy; another lies in its convenience for encoding problems from artificial intelligence and beyond. 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 [10] 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 taking only standard assumptions in computational complexity into account, i.e., assuming the Exponential Time Hypothesis (ETH). Intuitively, under ETH one cannot solve 3-SAT in time 2 o (n) where n is the number of variables. 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 second level of the polynomial hierarchy and a handful for the third level. 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) and thereby cover the full polynomial hierarchy. 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. By construction of the reduction, our results naturally carry over to the larger parameter pathwidth. As a by-product, our result renders the algorithm by Chen [10] asymptotically worst-case optimal.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 to foster research on lower bounds (parameterized by treewidth). Finally, we illustrate the usefulness of our results by discussing various applications of our results to problems that are located higher on the polynomial hierarchy, in particular, various problems from the literature such as projected model counting problems.

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Section: Methodology For Lower Boundsmentioning

confidence: 99%

Lower Bounds for QBFs of Bounded Treewidth

Fichte¹,

Hecher²,

Pfandler³

2019

Preprint

Self Cite

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show abstract

“…Proposition 12 (⋆, Fichte and Hecher [21]). DP PROJ runs in time O(2 4m · g · γ( F )) 6 where g is the number of nodes of the given TD of the underlying graph G F of the considered framework F and m := max{|ν(t)| | t ∈ N } for input TTD T purged = (T, χ, ν) of DP PROJ .…”

Section: Algorithms For Projected Credulous Counting By Exploiting Trmentioning

confidence: 99%

“…Hypothesis 15 (3ETH, Fichte and Hecher [21]). The problem ∃∀∃-SAT for a quantified Boolean formula Φ of treewidth k can not be decided in time…”

Section: Algorithms For Projected Credulous Counting By Exploiting Trmentioning

confidence: 99%

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Counting Complexity for Reasoning in Abstract Argumentation

Fichte

Hecher

Meier

2019

AAAI

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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.

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“…In order to deal with this high complexity efficiently, we propose to use a method from the field of parameterized complexity, namely, investigate how the runtime behaves when looking at different structural parameters of the problem. For standard ASP, this topic has received considerable interest, [3,13,14,18,25,31]. However, the parameterized complexity of epistemic ASP has remained largely unexplored so far.…”

Section: Introductionmentioning

confidence: 99%

Structural Decompositions of Epistemic Logic Programs

Hecher¹,

Morak²,

Woltran³

2020

Preprint

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Epistemic logic programs (ELPs) are a popular generalization of standard Answer Set Programming (ASP) providing means for reasoning over answer sets within the language. This richer formalism comes at the price of higher computational complexity reaching up to the fourth level of the polynomial hierarchy. However, in contrast to standard ASP, dedicated investigations towards tractability have not been undertaken yet. In this paper, we give first results in this direction and show that central ELP problems can be solved in linear time for ELPs exhibiting structural properties in terms of bounded treewidth. We also provide a full dynamic programming algorithm that adheres to these bounds. Finally, we show that applying treewidth to a novel dependency structure-given in terms of epistemic literals-allows to bound the number of ASP solver calls in typical ELP solving procedures.

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Treewidth and Counting Projected Answer Sets

Cited by 19 publications

References 33 publications

Lower Bounds for QBFs of Bounded Treewidth

Lower Bounds for QBFs of Bounded Treewidth

Counting Complexity for Reasoning in Abstract Argumentation

Structural Decompositions of Epistemic Logic Programs

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