Ronald de Haan

Fixed-Parameter Tractable Reductions to SAT

Today’s SAT solvers have an enormous importance and impact in many practical settings. They are used as efficient back-end to solve many NP-complete problems. However, many computational problems are located at the second level of the Polynomial Hierarchy or even higher, and hence polynomial-time transformations to SAT are not possible, unless the hierarchy collapses. In certain cases one can break through these complexity barriers by fixed-parameter tractable (fpt) reductions which exploit structural aspects of problem instances in terms of problem parameters. Recent research established a general theoretical framework that supports the classification of parameterized problems on whether they admit such an fpt-reduction to SAT or not. We use this framework to analyze some problems that are related to Boolean satisfiability. We consider several natural parameterizations of these problems, and we identify for which of these an fpt-reduction to SAT is possible. The problems that we look at are related to minimizing an implicant of a DNF formula, minimizing a DNF formula, and satisfiability of quantified Boolean formulas.

On the subexponential-time complexity of CSP

A Constraint Satisfaction Problem (CSP) with n variables ranging over a domain of d values can be solved by brute-force in dn steps (omitting a polynomial factor). With a more careful approach, this trivial upper bound can be improved for certain natural restrictions of the CSP. In this paper we establish theoretical limits to such improvements, and draw a detailed landscape of the subexponential-time complexity of CSP. We first establish relations between the subexponential-time complexity of CSP and that of other problems, including CNF-SAT. We exploit this connection to provide tight characterizations of the subexponential-time complexity of CSP under common assumptions in complexity theory. For several natural CSP parameters, we obtain threshold functions that precisely dictate the subexponential-time complexity of CSP with respect to the parameters under consideration. Our analysis provides fundamental results indicating whether and when one can significantly improve on the brute-force search approach for solving CSP.

Machine Characterizations for Parameterized Complexity Classes Beyond Para-NP

Due to the remarkable power of modern SAT solvers, one can efficiently solve NP-complete problems in many practical settings by encoding them into SAT. However, many important problems in various areas of computer science lie beyond NP, and thus we cannot hope for polynomial-time encodings into SAT. Recent research proposed the use of fixed-parameter tractable (fpt) reductions to provide efficient SAT encodings for these harder problems. The parameterized complexity classes ∃ k ∀ * and ∀ k ∃ * provide strong theoretical evidence that certain parameterized problems are not fpt-reducible to SAT. Originally, these complexity classes were defined via weighted satisfiability problems for quantified Boolean formulas, extending the general idea for the canonical problems for the Weft Hierarchy.

Stable Matching with Uncertain Linear Preferences

We consider the two-sided stable matching setting in which there may be uncertainty about the agents’ preferences due to limited information or communication. We consider three models of uncertainty: (1) lottery model — in which for each agent, there is a probability distribution over linear preferences, (2) compact indifference model — for each agent, a weak preference order is specified and each linear order compatible with the weak order is equally likely and (3) joint probability model — there is a lottery over preference profiles. For each of the models, we study the computational complexity of computing the stability probability of a given matching as well as finding a matching with the highest probability of being stable. We also examine more restricted problems such as deciding whether a certainly stable matching exists. We find a rich complexity landscape for these problems, indicating that the form uncertainty takes is significant.

Local backbones

A backbone of a propositional CNF formula is a variable whose truth value is the same in every truth assignment that satisfies the formula. The notion of backbones for CNF formulas has been studied in various contexts. In this paper, we introduce local variants of backbones, and study the computational complexity of detecting them. In particular, we consider k- backbones, which are backbones for sub-formulas consisting of at most k clauses, and iterative k- backbones, which are backbones that result after repeated instantiations of k- backbones. We determine the parameterized complexity of deciding whether a variable is a k- backbone or an iterative k- backbone for various restricted formula classes, including Horn, definite Horn, and Krom. We also present some first empirical results regarding backbones for CNF-Satisfiability (SAT). The empirical results we obtain show that a large fraction of the backbones of structured SAT instances are local, in contrast to random instances, which appear to have few local backbones.

Parameterized Complexity Classes Beyond Para-NP

Todays propositional satisfiability (SAT) solvers are extremely powerful and can be used as an efficient back-end for solving NP-complete problems. However, many fundamental problems in logic, in knowledge representation and reasoning, and in artificial intelligence are located at the second level of the Polynomial Hierarchy or even higher, and hence for these problems polynomial-time transformations to SAT are not possible, unless the hierarchy collapses. Recent research shows that in certain cases one can break through these complexity barriers by fixed-parameter tractable (fpt) reductions to SAT which exploit structural aspects of problem instances in terms of problem parameters. These reductions are more powerful because their running times can grow superpolynomially in the problem parameters. In this paper we develop a general theoretical framework that supports the classification of parameterized problems on whether they admit such an fpt-reduction to SAT or not.

A Dichotomy Result for Ramsey Quantifiers

Ramsey quantifiers are a natural object of study not only for logic and computer science, but also for formal semantics of natural language. Restricting attention to finite models leads to the natural question whether all Ramsey quantifiers are either polynomial-time computable or NP-hard, and whether we can give a natural characterization of the polynomial-time computable quantifiers. In this paper, we first show that there exist intermediate Ramsey quantifiers and then we prove a dichotomy result for a large and natural class of Ramsey quantifiers, based on a reasonable and widely-believed complexity assumption. We show that the polynomial-time computable quantifiers in this class are exactly the constant-log-bounded Ramsey quantifiers.

Subexponential Time Complexity of CSP with Global Constraints

Not all NP-complete problems share the same practical hardness with respect to exact computation. Whereas some NP-complete problems are amenable to efficient computational methods, others are yet to show any such sign. It becomes a major challenge to develop a theoretical framework that is more fine-grained than the theory of NP-completeness, and that can explain the distinction between the exact complexities of various NP-complete problems. This distinction is highly relevant for constraint satisfaction problems under natural restrictions, where various shades of hardness can be observed in practice.

Small Unsatisfiable Subsets in Constraint Satisfaction

The problem of finding small unsatisfiable subsets of a set of constraints is important for various applications in computer science and artificial intelligence. We study the problem of identifying whether a given instance to the constraint satisfaction problem (CSP) has an unsatisfiable subset of size at most k from a parameterized complexity point of view. We show that the problem of finding small unsatisfiable subsets of a CSP instance is harder than the corresponding problem for CNF formulas. Moreover, we show that the problem is not fixed-parameter tractable when restricting the problem to any maximal tractable Boolean constraint language (for which the problem is nontrivial). We show that the problem is hard even when the maximum number of occurrences of any variable is bounded by a constant, a restriction which leads to fixed-parameter tractability for the case of CNF formulas. Finally, we relate the problem of finding small unsatisfiable subsets to the problem of identifying variable assignments that are enforced already by a small number of constraints (backbones), or that are ruled out already by a small number of constraints (anti-backbones).

The regulargcc matrix constraint

We study propagation of a global constraint that ensures that each row of a matrix of decision variables satisfies a Regular constraint, and each column satisfies a Gcc constraint. On the negative side, we prove that propagation is NP-hard even under some strong restrictions (e.g. just 2 values, just 4 states in the automaton, just 5 columns to the matrix, or restricting to limited classes of automata). We also prove that propagation is W[2]-hard when the problem is parameterized by the number of rows in the matrix. On the positive side, we identify several cases where propagation is fixed parameter tractable.