Structure-Aware Lower Bounds and Broadening the Horizon of Tractability for QBF
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 PSPACE-complete problem. As such, it has previously been studied under various structural...
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| Veröffentlicht in: | 2023 38TH ANNUAL ACM/IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE, LICS S. 1 - 14 |
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IEEE
26.06.2023
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| Schriftenreihe: | IEEE Symposium on Logic in Computer Science |
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| ISBN: | 9798350335880, 9798350335873 |
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| Abstract | 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 PSPACE-complete 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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| AbstractList | 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 PSPACE-complete 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 problems 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. 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 PSPACE-complete 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. |
| Author | Ordyniak, Sebastian Ganian, Robert Slivovsky, Friedrich Hecher, Markus Fichte, Johannes K. |
| Author_xml | – sequence: 1 givenname: Johannes K. surname: Fichte fullname: Fichte, Johannes K. organization: Linköping University,Department of Computer and Information Science (IDA),Linköping,Sweden,SE-581 83 – sequence: 2 givenname: Robert surname: Ganian fullname: Ganian, Robert organization: Institute of Logic and Computation, TU Wien,Wien,Austria,1040 – sequence: 3 givenname: Markus surname: Hecher fullname: Hecher, Markus organization: Massachusetts Institute of Technology,Cambridge,MA,USA,02139 – sequence: 4 givenname: Friedrich surname: Slivovsky fullname: Slivovsky, Friedrich organization: Institute of Logic and Computation, TU Wien,Wien,Austria,1040 – sequence: 5 givenname: Sebastian surname: Ordyniak fullname: Ordyniak, Sebastian organization: University of Leeds,School of Computing,Leeds,United Kingdom,LS2 9JT |
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| SubjectTerms | Complexity theory Computer science Filling Probabilistic logic Structural engineering Upper bound |
| Title | Structure-Aware Lower Bounds and Broadening the Horizon of Tractability for QBF |
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