Bounded-width QBF is PSPACE-complete

Atserias, Albert; Oliva, Sergi

doi:10.1016/j.jcss.2014.04.014

Cited by 24 publications

(37 citation statements)

References 8 publications

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“…After showing that QBF remains PSPACE-complete on graphs of bounded pathwidth [3], the authors introduced a width parameter based on treewidth, which is called respectful treewidth, that allows to take into account dependencies between the variables in a QBF formula. They showed that QBF is fixed-parameter tractable parameterized by respectful treewidth provided that a corresponding tree decomposition is given as part of the input.…”

Section: Related Workmentioning

confidence: 99%

“…Prominent examples of decompositions used in such techniques include decompositions for the structural parameters treewidth [30], pathwidth [29], clique-width [10] and rankwidth [25]; all of these are known to support FPT algorithms for SAT [38,19], but the same is not true for QBF. Indeed QBF remains PSPACE-complete even on instances with constant pathwidth [3]. As a consequence, many classes of QBFs that have a natural and seemingly "simple" structure remained beyond the reach of current algorithmic techniques; this is also witnessed by previous work of Pan and Vardi [26] that established strong lower bounds for the problem.…”

Section: Introductionmentioning

confidence: 99%

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Using decomposition-parameters for QBF: Mind the prefix!

Eiben

Ganian

Ordyniak

2020

Journal of Computer and System Sciences

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Similar to the satisfiability (SAT) problem, which can be seen to be the archetypical problem for NP, the quantified Boolean formula problem (QBF) is the archetypical problem for PSPACE. Recently, Atserias and Oliva (2014) showed that, unlike for SAT, many of the well-known decompositional parameters (such as treewidth and pathwidth) do not allow efficient algorithms for QBF. The main reason for this seems to be the lack of awareness of these parameters towards the dependencies between variables of a QBF formula. In this paper we extend the ordinary pathwidth to the QBF-setting by introducing prefix pathwidth, which takes into account the dependencies between variables in a QBF, and show that it leads to an efficient algorithm for QBF. We hope that our approach will help to initiate the study of novel tailor-made decompositional parameters for QBF and thereby help to lift the success of these decompositional parameters from SAT to QBF.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Using decomposition-parameters for QBF: Mind the prefix!

Eiben

Ganian

Ordyniak

2020

Journal of Computer and System Sciences

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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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“…Atserias and Oliva [1]) in order to achieve bounded treewidth of the graph used. However, Atserias and Oliva [1] note that bounded treewidth quantified boolean formula become polynomial time solvable when restricted to respectful bounded treewidth instances by a result of Chen and Dalmau [3]. Thus, a potential PSPACE-completeness result for Shortest Path Game on bounded treewidth graphs will likely require a different approach; the computational complexity of Shortest Path Game on bounded treewidth graphs remains an interesting open question.…”

Section: More General Classes Of Graphs and Some Observationsmentioning

confidence: 99%

On the Shortest Path Game

Darmann

Pferschy

Schauer

2017

Discrete Applied Mathematics

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In this work we address a game theoretic variant of the shortest path problem, in which two decision makers (players) move together along the edges of a graph from a given starting vertex to a given destination. The two players take turns in deciding in each vertex which edge to traverse next. The decider in each vertex also has to pay the cost of the chosen edge. We want to determine the path where each player minimizes its costs taking into account that also the other player acts in a selfish and rational way. Such a solution is a subgame perfect equilibrium and can be determined by backward induction in the game tree of the associated finite game in extensive form.We show that the decision problem associated with such a path is PSPACEcomplete even for bipartite graphs both for the directed and the undirected version. The latter result is a surprising deviation from the complexity status of the closely related game Geography.On the other hand, we can give polynomial time algorithms for directed acyclic graphs and for cactus graphs even in the undirected case. The latter is based on a decomposition of the graph into components and their resolution by a number of fairly involved dynamic programming arrays. Finally, we give some arguments about closing the gap of the complexity status for graphs of bounded treewidth.

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Bounded-width QBF is PSPACE-complete

Cited by 24 publications

References 8 publications

Using decomposition-parameters for QBF: Mind the prefix!

Using decomposition-parameters for QBF: Mind the prefix!

QBF as an Alternative to Courcelle’s Theorem

On the Shortest Path Game

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