Treewidth-Aware Quantifier Elimination and Expansion for QCSP

Fichte, Johannes Klaus; Hecher, Markus; I., Kieler, Maximilian F.

doi:10.1007/978-3-030-58475-7_15

Cited by 4 publications

(1 citation statement)

References 33 publications

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“…Consequently, one obtains similar results as in related work [Lampis and Mitsou, 2017], but for quantifier rank ≥ 3. Indeed, under the ETH, the validity of QCSPs Q over domain D of quantifier rank , where [Fichte et al, 2020b].…”

Section: Corollary 515 (Pathwidth Bound) Given An Arbitrary Qbf Of Th...mentioning

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: Corollary 515 (Pathwidth Bound) Given An Arbitrary Qbf Of Th...mentioning

confidence: 99%

Advanced tools and methods for treewidth-based problem solving

Hecher¹

2023

It - Information Technology

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A Practical Account into Counting Dung’s Extensions by Dynamic Programming

Dewoprabowo

Fichte²,

Gorczyca³

et al. 2022

Logic Programming and Nonmonotonic Reasoning

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Structure-Aware Lower Bounds and Broadening the Horizon of Tractability for QBF

Fichte

Ganian

Hecher

et al. 2023

2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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The QSAT problem, which asks to evaluate a quantified Boolean formula (QBF), is of fundamental interest in approximation, counting, decision, and probabilistic complexity and is also considered the prototypical PSPACEcomplete problem. As such, it has previously been studied under various structural restrictions (parameters), most notably parameterizations of the primal graph representation of instances. Indeed, it is known that QSAT remains PSPACE-complete even when restricted to instances with constant treewidth of the primal graph, but the problem admits a double-exponential fixed-parameter algorithm parameterized by the vertex cover number (primal graph).However, prior works have left a gap in our understanding of the complexity of QSAT when viewed from the perspective of other natural representations of instances, most notably via incidence graphs. In this paper, we develop structure-aware reductions which allow us to obtain essentially tight lower bounds for highly restricted instances of QSAT, including instances whose incidence graphs have bounded treedepth or feedback vertex number. We complement these lower bounds with novel algorithms for QSAT which establish a nearly-complete picture of the problem's complexity under standard graph-theoretic parameterizations. We also show implications for other natural graph representations, and obtain novel upper as well as lower bounds for QSAT under more fine-grained parameterizations of the primal graph.

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Treewidth-Aware Quantifier Elimination and Expansion for QCSP

Cited by 4 publications

References 33 publications

Advanced tools and methods for treewidth-based problem solving

Advanced tools and methods for treewidth-based problem solving

A Practical Account into Counting Dung’s Extensions by Dynamic Programming

Structure-Aware Lower Bounds and Broadening the Horizon of Tractability for QBF

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