Using decomposition-parameters for QBF: Mind the prefix!

Eiben, Eduard; Ganian, Robert; Ordyniak, Sebastian

doi:10.1016/j.jcss.2019.12.005

Cited by 6 publications

(6 citation statements)

References 25 publications

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“…Recall that indeed, treewidth has been widely employed for applications such as Boolean satisfiability (Sat) [Samer and Szeider, 2010] and constraint satisfaction (CSP) [Dechter, 2006;Freuder, 1985], but also for problems believed beyond NP such as probabilistic inference [Ordyniak and Szeider, 2013] as well as problems in knowledge representation and reasoning [Gottlob et al, 2010;Pichler et al, 2010;Dvořák et al, 2012]. Also for the prominent problem QSat, which asks for deciding the validity of a quantified Boolean formula (QBF), there are tractability results using an additional parameter [Chen, 2004] or some extension of treewidth [Eiben et al, 2018[Eiben et al, , 2020.…”

Section: Lower Bounds By Decomposition-guided Reductionsmentioning

confidence: 99%

Advanced tools and methods for treewidth-based problem solving

Hecher¹

2023

It - Information Technology

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Computer programs, so-called solvers, for solving the well-known Boolean satisfiability problem (Sat) have been improving for decades. Among the reasons, why these solvers are so fast, is the implicit usage of the formula’s structural properties during solving. One of such structural indicators is the so-called treewidth, which tries to measure how close a formula instance is to being easy (tree-like). This work focuses on logic-based problems and treewidth-based methods and tools for solving them. Many of these problems are also relevant for knowledge representation and reasoning (KR) as well as artificial intelligence (AI) in general. We present a new type of problem reduction, which is referred to by decomposition-guided (DG). This reduction type forms the basis to solve a problem for quantified Boolean formulas (QBFs) of bounded treewidth that has been open since 2004. The solution of this problem then gives rise to a new methodology for proving precise lower bounds for a range of further formalisms in logic, KR, and AI. Despite the established lower bounds, we implement an algorithm for solving extensions of Sat efficiently, by directly using treewidth. Our implementation is based on finding abstractions of instances, which are then incrementally refined in the process. Thereby, our observations confirm that treewidth is an important measure that should be considered in the design of modern solvers.

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Section: Lower Bounds By Decomposition-guided Reductionsmentioning

confidence: 99%

Advanced tools and methods for treewidth-based problem solving

Hecher¹

2023

It - Information Technology

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“…We remark that general QBF of bounded treewidth without any restriction on the quantifier prefix is PSPACE-complete [3], and finding tractable fragments by taking into account the structure of the prefix and notions similar to treewidth is quite an active area of research, see e.g. [15,14].…”

Section: Corollary 32mentioning

confidence: 99%

QBF as an Alternative to Courcelle’s Theorem

Lampis¹,

Mengel²,

Mitsou³

2018

Theory and Applications of Satisfiability Testing – SAT 2018

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We propose reductions to quantified Boolean formulas (QBF) as a new approach to showing fixed-parameter linear algorithms for problems parameterized by treewidth. We demonstrate the feasibility of this approach by giving new algorithms for several well-known problems from artificial intelligence that are in general complete for the second level of the polynomial hierarchy. By reduction from QBF we show that all resulting algorithms are essentially optimal in their dependence on the treewidth. Most of the problems that we consider were already known to be fixed-parameter linear by using Courcelle's Theorem or dynamic programming, but we argue that our approach has clear advantages over these techniques: on the one hand, in contrast to Courcelle's Theorem, we get concrete and tight guarantees for the runtime dependence on the treewidth. On the other hand, we avoid tedious dynamic programming and, after showing some normalization results for CNF-formulas, our upper bounds often boil down to a few lines.

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“…In this paper, we initiate the investigation of DQBF through the lens of parameterized complexity by developing the first two fixed-parameter algorithms for the problem. As for the choice of parameters, we follow up on previous successful work on SAT (Samer and Szeider 2009b;Ganian and Szeider 2017), QBF (Chen 2004;Atserias and Oliva 2014;Eiben, Ganian, and Ordyniak 2018;Eiben, Ganian, and Ordyniak 2020) as well as numerous other problems (Ganian and Ordyniak 2018;Samer and Szeider 2010) by considering restrictions to the primal graph 2 -a natural graph representation which captures how variables in the instance interact with each other. The by far most prevalent restriction in this context is the parameterization by treewidth, a graph parameter which measures how tree-like the graph is, and our first result aims at establishing the tractability of DQBF evaluation with respect to the treewidth of the primal graph (i.e., the primal treewidth).…”

Section: Introductionmentioning

confidence: 99%

“…The more recent of these, called dependency treewidth (Eiben, Ganian, and Ordyniak 2018), is designed in a way which allows instances of small width to be solved by Q-resolution (Kleine Büning, Karpinski, and Flögel 1995)-a proof system for QBF which does not have any directly applicable equivalent in DQBF. The other result is centered around prefix treewidth (Eiben, Ganian, and Ordyniak 2020), which is designed in a way that facilitates the construction of a model by dynamic programming. While on a high level such an approach might seem fruitful also for DQBF evaluation, due to the lack of a linear quantifier order in DQBF it is not clear at all whether or how techniques from that paper can be adapted to our more general setting.…”

Section: Introductionmentioning

confidence: 99%

Fixed-Parameter Tractability of Dependency QBF with Structural Parameters

Ganian

Peitl

Slivovsky

et al. 2020

Proceedings of the Seventeenth International Conference on Principles of Knowledge Representation and Reasoning

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We study dependency quantified Boolean formulas (DQBF), an extension of QBF in which dependencies of existential variables are listed explicitly rather than being implicit in the order of quantifiers. DQBF evaluation is a canonical NEXPTIME-complete problem, a complexity class containing many prominent problems that arise in Knowledge Representation and Reasoning. One approach for solving such hard problems is to identify and exploit structural properties captured by numerical parameters such that bounding these parameters gives rise to an efficient algorithm. This idea is captured by the notion of fixed-parameter tractability (FPT). We initiate the study of DQBF through the lens of fixed-parameter tractability and show that the evaluation problem becomes FPT under two natural parameterizations: the treewidth of the primal graph of the DQBF instance combined with a restriction on the interactions between the dependency sets, and also the treedepth of the primal graph augmented by edges representing dependency sets.

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Using decomposition-parameters for QBF: Mind the prefix!

Cited by 6 publications

References 25 publications

Advanced tools and methods for treewidth-based problem solving

Advanced tools and methods for treewidth-based problem solving

QBF as an Alternative to Courcelle’s Theorem

Fixed-Parameter Tractability of Dependency QBF with Structural Parameters

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