Friedrich Slivovsky

Soundness of Q-resolution with dependency schemes

Q-resolution and Q-term resolution are proof systems for quantified Boolean formulas (QBFs). We introduce generalizations of these proof systems named Q ( D ) -resolution and Q ( D ) -term resolution. Q ( D ) -resolution and Q ( D ) -term resolution are parameterized by a dependency scheme D and use more powerful ?-reduction and ?-reduction rules, respectively. We show soundness of these systems for particular dependency schemes: we prove (1) soundness of Q ( D ) -resolution parameterized by the reflexive resolution-path dependency scheme, and (2) soundness of Q ( D ) -term resolution parameterized by the resolution-path dependency scheme. These results entail soundness of the proof systems used for certificate generation in the state-of-the-art solver DepQBF.

Model Counting for CNF Formulas of Bounded Modular Treewidth

We define the modular treewidth of a graph as its treewidth after contraction of modules. This parameter properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments can be computed in polynomial time for CNF formulas whose incidence graphs have bounded modular treewidth. Our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial bounding the runtime of our algorithm depends on the modular treewidth of the input formula. We show that it is unlikely that this dependency can be avoided by proving that SAT is W[1]-hard when parameterized by the modular incidence treewidth of the given CNF formula.

Computing resolution-path dependencies in linear time ,

The alternation of existential and universal quantifiers in a quantified boolean formula (QBF) generates dependencies among variables that must be respected when evaluating the formula. Dependency schemes provide a general framework for representing such dependencies. Since it is generally intractable to determine dependencies exactly, a set of potential dependencies is computed instead, which may include false positives. Among the schemes proposed so far, resolution path dependencies introduce the fewest spurious dependencies. In this work, we describe an algorithm that detects resolution-path dependencies in linear time, resolving a problem posed by Van Gelder (CP 2011).

Variable Dependencies and Q-Resolution

We propose Q(D)-resolution, a proof system for Quantified Boolean Formulas. Q(D)-resolution is a generalization of Q-resolution parameterized by a dependency scheme D. This system is motivated by the generalization of the QDPLL algorithm using dependency schemes implemented in the solver DepQBF. We prove soundness of Q(D)-resolution for a dependency scheme D that is strictly more general than the standard dependency scheme; the latter is currently used by DepQBF. This result is obtained by proving correctness of an algorithm that transforms Q(D)-resolution refutations into Q-resolution refutations and could be of independent practical interest. We also give an alternative characterization of resolution- path dependencies in terms of directed walks in a formula’s implication graph which admits an algorithmically more advantageous treatment.

On Compiling CNFs into Structured Deterministic DNNFs

We show that the traces of recently introduced dynamic programming algorithms for #SAT can be used to construct structured deterministic DNNF (decomposable negation normal form) representations of propositional formulas in CNF (conjunctive normal form). This allows us prove new upper bounds on the complexity of compiling CNF formulas into structured deterministic DNNFs in terms of parameters such as the treewidth and the clique-width of the incidence graph.

Model Counting for Formulas of Bounded Clique-Width

We show that #SAT is polynomial-time tractable for classes of CNF formulas whose incidence graphs have bounded symmetric clique-width (or bounded clique-width, or bounded rank-width). This result strictly generalizes polynomial-time tractability results for classes of formulas with signed incidence graphs of bounded clique-width and classes of formulas with incidence graphs of bounded modular treewidth, which were the most general results of this kind known so far.

Dependency Learning for QBF

Quantified Boolean Formulas (QBFs) can be used to succinctly encode problems from domains such as formal verification, planning, and synthesis. One of the main approaches to QBF solving is Quantified Conflict Driven Clause Learning (QCDCL). By default, QCDCL assigns variables in the order of their appearance in the quantifier prefix so as to account for dependencies among variables. Dependency schemes can be used to relax this restriction and exploit independence among variables in certain cases, but only at the cost of nontrivial interferences with the proof system underlying QCDCL. We propose a new technique for exploiting variable independence within QCDCL that allows solvers to learn variable dependencies on the fly. The resulting version of QCDCL enjoys improved propagation and increased flexibility in choosing variables for branching while retaining ordinary (long-distance) Q-resolution as its underlying proof system. In experiments on standard benchmark sets, an implementation of this algorithm shows performance comparable to state-of-the-art QBF solvers.

Meta-kernelization with Structural Parameters

Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS’09) and of Fomin et al. (FOCS’10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use structural parameters of the input and not the solution size. Let \(\mathcal{C}\) be a graph class. We define the \(\mathcal{C}\) - cover number of a graph to be the smallest number of modules the vertex set can be partitioned into such that each module induces a subgraph that belongs to the class \(\mathcal{C}\).

Meta-kernelization with structural parameters

Kernelization is a polynomial-time algorithm that reduces an instance of a parameterized problem to a decision-equivalent instance, the kernel, whose size is bounded by a function of the parameter. In this paper we present meta-theorems that provide polynomial kernels for large classes of graph problems parameterized by a structural parameter of the input graph. Let C be an arbitrary but fixed class of graphs of bounded rank-width (or, equivalently, of bounded clique-width). We define the C -cover number of a graph to be the smallest number of modules its vertex set can be partitioned into, such that each module induces a subgraph that belongs to C . We show that each decision problem on graphs which is expressible in Monadic Second Order (MSO) logic has a polynomial kernel with a linear number of vertices when parameterized by the C -cover number. We provide similar results for MSO expressible optimization and modulo-counting problems. Kernelization meta-algorithms parameterized by structural graph parameters.Preprocessing for MSO definable decision, optimization and counting problems.Parameter based on a combination of rank-width and modular decompositions.

Quantifier Reordering for QBF

State-of-the-art procedures for evaluating quantified Boolean formulas often expect input formulas in prenex conjunctive normal form (PCNF). We study dependency schemes as a means of reordering the quantifier prefix of a PCNF formula while preserving its truth value. Dependency schemes associate each formula with a binary relation on its variables (the dependency relation) that imposes constraints on certain operations manipulating the formula’s quantifier prefix. We prove that known dependency schemes support a stronger reordering operation than was previously known. We present an algorithm that, given a formula and its dependency relation, computes a compatible reordering with a minimum number of quantifier alternations. In combination with a dependency scheme that can be computed in polynomial time, this yields a polynomial time heuristic for reducing the number of quantifier alternations of an input formula. The resolution-path dependency scheme is the most general dependency scheme introduced so far. Using an interpretation of resolution paths as directed paths in a formula’s implication graph, we prove that the resolution-path dependency relation can be computed in polynomial time.