Using decomposition-parameters for QBF: Mind the prefix!
Introduction
Many important computational tasks such as verification, planning, and several questions in knowledge representation and automated reasoning can be naturally encoded as the problem of evaluating quantified Boolean formulas [15], [24], [28], [31], a generalization of the propositional satisfiability problem (SAT). In recent years quantified Boolean formulas have become a very active research area. The problem of evaluating quantified Boolean formulas, called QBF, is the archetypical PSpace-complete problem and is therefore believed to be computationally harder than the NP-complete propositional satisfiability problem [21], [27], [37].
In spite of the close connection between QBF and SAT, many of the tools and techniques that work for SAT are not known to help for QBF, and this is especially true for so-called decomposition-based techniques [2]. Such techniques use various kinds of decompositions to capture the structure of the input, leading to efficient algorithms for computing solutions with run-time guarantees. Decomposition-based techniques are tied to a numerical parameter k, which represents the fitness of the decomposition. The goal is then to obtain algorithms whose running time is polynomial in the input size n and exponential only in k, i.e., with a running time of where f is some computable function; such algorithms are called FPT algorithms, and problems that admit an FPT algorithm w.r.t. some parameters belong to the class FPT. Prominent examples of decompositions used in such techniques include decompositions for the structural parameters treewidth [30], pathwidth [29], clique-width [10] and rank-width [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.
In this work we introduce and develop prefix pathwidth, which is a novel decomposition-based parameter that allows an FPT algorithm for QBF. Prefix pathwidth is an extension of pathwidth, which takes into account not only the structure of clauses in the formula, but also the structure contained in the quantification of variables. To achieve the latter, we make use of the dependency schemes introduced by Samer and Szeider ([32], [34]), see also the work of Biere and Lonsing ([6]). Dependency schemes capture how the assignment of individual variables in a QBF depends on other variables, and research in this direction has uncovered a large number of distinct dependency schemes. The most basic dependency scheme is called the trivial dependency scheme [32], which stipulates that each variable depends on all variables with distinct quantification that come before it in the prefix. When using this dependency scheme, we obtain (by combining Theorem 2 with Theorem 34):
Theorem 1 QBF is FPT parameterized by the prefix pathwidth with respect to the trivial dependency scheme.
All of our results can also be applied to prefix pathwidth with respect to so-called permutation dependency schemes [35]. Informally, a dependency scheme is a permutation dependency scheme if the satisfiability of a QBF formula is independent of the ordering of the variables in the quantifier prefix as long as the ordering is consistent with the ordering implied by the dependency scheme. Since almost all known dependency schemes are permutation dependency schemes all our results apply to a wide range of dependency schemes. In practice, using different dependency schemes may lead to better prefix path-decompositions, in turn resulting in significantly faster algorithms.
In their full generality, our main results on solving QBF using prefix pathwidth can be separated into two components:
- 1.
using a prefix path-decomposition of small prefix pathwidth to solve the given QBF I, and
- 2.
finding a suitable prefix path-decomposition to be used for step 1.
We resolve the first task by applying advanced dynamic programming techniques on partial existential strategies for the Hintikka game (see e.g., the work of Grädel et al. [14]) played on the QBF. Essentially, the game approach allows us to translate the question of whether a QBF is true to the question of whether there exists a winning strategy for one player in the Hintikka game. We show that although the number of such strategies is unbounded, at each point in the prefix path-decomposition there is only a small number of partial strategies on the processed vertices that need to be considered. Thus we obtain:
Theorem 2 QBF is FPT parameterized by the width of a prefix path-decomposition w.r.t. any permutation dependency scheme, when such a decomposition is provided as part of the input.
Resolving step 2 boils down to an algorithmic problem on graphs, which is related to the problem of computing various established parameters of directed graphs, such as directed pathwidth or directed treewidth. It is an important open problem whether computing these parameters is FPT or not [39] and the same obstacles seem to also be present for computing our parameter in the general sense. To bypass this barrier, we develop new algorithmic techniques to obtain three distinct algorithms for computing prefix path-decompositions. The first of these algorithms, presented in Theorem 34, works for the trivial dependency poset as well as other posets that have a similar “layered” structure. The latter two of our algorithms then focus on general posets, but their performance depends on the poset-width (i.e., the size of a maximum anti-chain) of the dependency relation; on a high level, the poset-width captures the density of dependencies between variables. In particular, we obtain one polynomial-time approximation algorithm (Theorem 30) and one FPT algorithm (Theorem 29). In combination with the previous Theorem 2, Theorem 30 yields one of our main contributions, formalized in Theorem 3 below. Observe that here we do not require a decomposition to be part of the input.
Theorem 3 Let τ be a fixed permutation dependency scheme. There exists an FPT algorithm that takes as input a QBF I and decides whether I is true in time , where f is a computable function, k is the prefix pathwidth and w is the poset-width of I w.r.t. τ.
We remark that our results have implications for the tractability of QBF with respect to already established structural parameters. We provide an example of this in the Concluding Notes, where we show that QBF is FPT when parameterized by the vertex cover number of the matrix (irrespective of the prefix), i.e., the QBF formula without the prefix.
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. Similar results have been shown for first-order model checking [1] and quantified constraint satisfaction [9]. In the former reference, it is also shown that computing an optimal respectful tree decomposition is fixed-parameter tractable, showing that QBF is fixed-parameter tractable parameterized by respectful treewidth. As we will show in Section 3, respectful treewidth is incomparable to our new parameter prefix pathwidth. Informally, there are two main differences between respectful treewidth and prefix pathwidth: (1) whereas respectful treewidth requires the ordering in which the variables are introduced (along the tree decomposition) to be compatible with the dependencies, prefix pathwidth needs the ordering in which the variables are forgotten (along the path decomposition) to be compatible with the dependencies and (2) respectful treewidth is solely defined for the trivial dependency scheme, while prefix pathwidth allows the use of arbitrary permutation dependency schemes. Other structural parameters such as backdoors have also been studied in the context of QBF [32].
Recent follow-up work that builds upon the results established in this paper include a paper by Lampis and Mitsou [23], which (1) investigates upper and lower bounds when using treewidth to solve QBF with a single quantifier alternation, and (2) improves the running time of the fixed-parameter algorithm parameterized by the vertex cover number presented in our Theorem 40. In another recent follow-up [16], the authors of this paper investigate a different and incomparable parameter related to treewidth that can be used for solving QBF; the incomparability of that parameter with the one presented here is established in Lemma 4 in [16].
Section snippets
Preliminaries
For , we let denote the set . We refer to the book by Diestel ([12]) for standard graph-theoretic terminology. Given an undirected graph G, we denote by and its vertex and edge set, respectively. We use ab as a shorthand for the edge . For a set of vertices the guards of , denoted by , are the vertices in with at least one neighbor in . For a vertex , we denote by the set of its neighbors (excluding v) and for a vertex set
Prefix pathwidth for QBF
Let be a graph and be a partial order of V. For a vertex , we denote by the downward closure of v w.r.t. , i.e., the set . Similarly, for we let .
Let be a tree decomposition of G. For a node t of T we denote by the subtree of T rooted at t, by the set , and by the set . For a vertex we denote by the unique node t satisfying and , where s is the parent of t in T. For a path
Using prefix pathwidth
In this section we will show that deciding the satisfiability of a QBF is fixed-parameter tractable parameterized by the width of a prefix path-decomposition, which is assumed to be provided as part of the input. The next section will then show how such a prefix path-decomposition can be computed efficiently.
Computing prefix pathwidth
This section is devoted to parameterized and approximation algorithms for computing the prefix pathwidth. Observe that the prefix pathwidth of the graph G w.r.t. the empty partial ordering is the same as the pathwidth of G. Therefore, since computing pathwidth is NP-complete so is computing prefix pathwidth.
Before we present our algorithms, we will state some interesting observations about prefix path-decompositions. For the remainder of this section let G be a graph and a poset on
Concluding notes
Our results push the frontiers of tractability for QBF to new natural classes of instances. We provide one specific example of this below. A vertex cover of a graph G is a vertex set of G which is incident to each edge in G, and the vertex cover number of G is the minimum size of a vertex cover in G. The vertex cover number has often been used as a structural parameter for graph problems which do not have FPT algorithms parameterized by treewidth (see for instance [17]).
Theorem 40 QBF is fixed parameter
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
Robert Ganian was supported by the Austrian Science Fund (FWF, Project P31336).
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