Parameterised Enumeration for Modification Problems

Creignou, Nadia; Ktari, Raïda; Meier, Arne; Müller, Julian-Steffen; Olive, Frédéric; Vollmer, Heribert

doi:10.3390/a12090189

Cited by 14 publications

(5 citation statements)

References 39 publications

(58 reference statements)

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“…Since argumentation has been applied to questions in human reasoning (Dietz, Kakas, & Michael, 2021 and quantitative reasoning plays an interesting role there as well (Dietz, Fichte, & Hamiti, 2022), we suspect that counting and probabilistic questions can have an interesting application there as well. Considering the (parameterized) enumeration complexity (Johnson, Papadimitriou, & Yannakakis, 1988;Creignou, Meier, Müller, Schmidt, & Vollmer, 2017;Creignou, Ktari, Meier, Müller, Olive, & Vollmer, 2019;Meier, 2020) of the studied problems is also planned as future work. Finally, other measures such as fractional hypertree width or backdoors (Dvořák, Hecher, König, Schidler, Szeider, & Woltran, 2022) might be interesting to consider.…”

Section: Discussionmentioning

confidence: 99%

Counting Complexity for Reasoning in Abstract Argumentation

Fichte,

Hecher,

Meier

2024

jair

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In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics, including credulous reasoning. 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 the 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. Our algorithms run in double or triple exponential time in the treewidth, depending on the semantics under consideration. Finally, we establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension under the exponential time hypothesis (ETH).

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Section: Discussionmentioning

confidence: 99%

Counting Complexity for Reasoning in Abstract Argumentation

Fichte,

Hecher,

Meier

2024

jair

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“…A further research direction is to study our algorithm for practical applications and stronger parameters, e.g., (Grohe and Marx 2014;Greco et al 2018), particularly, in the light of recent decomposition techniques (Gottlob, Okulmus, and Pichler 2020). Also enumeration complexity (Creignou et al 2017(Creignou et al , 2019Meier 2020) is worth studying in this context.…”

Section: Discussionmentioning

confidence: 99%

Knowledge-Base Degrees of Inconsistency: Complexity and Counting

Fichte

Hecher

Meier

2021

AAAI

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Description logics (DLs) are knowledge representation languages that are used in the field of artificial intelligence (AI). A common technique is to query DL knowledge bases, e.g., by Boolean Datalog queries, and ask for entailment. But real world knowledge-bases are often obtained by combining data from various sources. This, inherently, might result in certain inconsistencies (with respect to a given query) and requires to estimate a degree of inconsistency before using a knowledge-base. In this paper, we provide a complexity analysis of fixed-domain non-entailment (NE) on Datalog programs for well-established families of knowledge bases (KBs). We exhibit a detailed complexity map for the decision cases, counting and projected counting, which may serve as a quantitative measure for inconsistency of a KB with respect to a query. Our results show that NE is natural for the second, third, and fourth level of the polynomial (counting) hierarchy depending on the type of the studied query (stratified, normal, disjunctive) and one level higher for the projected versions. Further, we show fixed-parameter tractability by bounding the treewidth, provide a constructive algorithm, and show its theoretical limitation in terms of conditional lower bounds.

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“…As further future work, we plan investigating the (parameterized) enumeration complexity [19,13,12,27] of reasoning in this setting.…”

Section: Discussionmentioning

confidence: 99%

Parameterized Complexity of Logic-Based Argumentation in Schaefer's Framework

Mahmood¹,

Meier²,

Schmidt³

2021

Preprint

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Logic-based argumentation is a well-established formalism modelling nonmonotonic reasoning. It has been playing a major role in AI for decades, now. Informally, a set of formulas is the support for a given claim if it is consistent, subset-minimal, and implies the claim. In such a case, the pair of the support and the claim together is called an argument. In this paper, we study the propositional variants of the following three computational tasks studied in argumentation: ARG (exists a support for a given claim with respect to a given set of formulas), ARG-Check (is a given set a support for a given claim), and ARG-Rel (similarly as ARG plus requiring an additionally given formula to be contained in the support). ARG-Check is complete for the complexity class DP, and the other two problems are known to be complete for the second level of the polynomial hierarchy (Parson et al., J. Log. Comput., 2003) and, accordingly, are highly intractable. Analyzing the reason for this intractability, we perform a two-dimensional classification: first, we consider all possible propositional fragments of the problem within Schaefer's framework (STOC 1978), and then study different parameterizations for each of the fragment. We identify a list of reasonable structural parameters (size of the claim, support, knowledge-base) that are connected to the aforementioned decision problems. Eventually, we thoroughly draw a fine border of parameterized intractability for each of the problems showing where the problems are fixed-parameter tractable and when this exactly stops. Surprisingly, several cases are of very high intractability (paraNP and beyond).

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Parameterised Enumeration for Modification Problems

Cited by 14 publications

References 39 publications

Counting Complexity for Reasoning in Abstract Argumentation

Counting Complexity for Reasoning in Abstract Argumentation

Knowledge-Base Degrees of Inconsistency: Complexity and Counting

Parameterized Complexity of Logic-Based Argumentation in Schaefer's Framework

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