Parameterized complexity classes beyond para-NP

Abstract

Today's 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 knowledge representation and reasoning 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. Recent research shows that 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.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. This framework is based on several new parameterized complexity classes. As a running example, we use the framework to classify the complexity of the consistency problem for disjunctive answer set programming, with respect to various natural parameters. We underpin the robustness of our theory by providing a characterization of the new complexity classes in terms of weighted QBF satisfiability, alternating Turing machines, and first-order model checking. In addition, we provide a compendium of parameterized problems that are complete for the new complexity classes, including problems related to Knowledge Representation and Reasoning, Logic, and Combinatorics. * This paper contains results that have appeared in shortened and preliminary form in the proceedings of SAT 2014 [40], the proceedings of KR 2014 [41], and the proceedings of SOFSEM 2015 [42].

A Compendium of Parameterized Problems at Higher Levels of the Polynomial Hierarchy

“…However, for such problems, there still could exist SAT encodings which can be produced in fpt-time with respect to some parameter associated with the problem. In fact, such fpt-time SAT encodings have been obtained for various problems on the second level of the PH [28][29][30][31]. The classes para-NP and para-co-NP contain exactly those parameterized problems that admit such a many-one fpt-reduction to SAT 1 and UNSAT 1 , respectively.…”

“…There are problems that apparently do not admit fpt-time encodings to SAT, but seem not to be para-Σ p 2 -hard nor para-Π p 2 -hard either. Recently, several complexity classes have been introduced to classify such intermediate problems [7,8,30]. These parameterized complexity classes are dubbed the k- * class and the * -k hierarchy, inspired by their definition, which is based on the following weighted variants of the quantified Boolean satisfiability problem that is canonical for the second level of the PH.…”

“…(See also Figure 1 for a visual overview of these inclusion relations.) Dual to the classical complexity class Σ [7,8]. Similarly, the classes Σ [7,8].…”

“…This indicates that, to establish whether an fpt-reduction to SAT is possible, a suitable parameterized complexity analysis at higher levels of the Polynomial Hierarchy is needed. Various parameterized complexity classes have been developed for this purpose [7,8] that, together with previously studied classes, enable such analyses. This includes both classes that correspond to various types of fpt-reductions to SAT-e.g., para-NP, para-co-NP, and FPT NP [few]-and classes that indicate that no fpt-reductions to SAT are possible, under suitable complexity-theoretic assumptions-e.g., Σ…”

Differential Privacy via Weighted Sampling Set Cover

“…Weighted set cover problem is the algorithmic frameworks of our methods, also a classic NP-problem [14] [15]. The main issue of this problem is how to find the approximation subset sets in each iteration, a good subsets can reduce the time complexity of the algorithm.…”