Parameterized complexity classes beyond para-NP

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 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.

[1]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[2]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[3]  Michael R. Fellows,et al.  Fixed-Parameter Tractability and Completeness II: On Completeness for W[1] , 1995, Theor. Comput. Sci..

[4]  Michael R. Fellows,et al.  Review of: Fundamentals of Parameterized Complexity by Rodney G. Downey and Michael R. Fellows , 2015, SIGA.

[5]  Michael Gelfond,et al.  Classical negation in logic programs and disjunctive databases , 1991, New Generation Computing.

[6]  Georg Gottlob,et al.  Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence Robust Constraint Satisfaction and Local Hidden Variables in Quantum Mechanics , 2022 .

[7]  Oded Goldreich.  P, NP, and NP-Completeness: The Basics of Complexity Theory , 2010 .

[8]  Ronald de Haan,et al.  Fixed-Parameter Tractable Reductions to SAT for Planning , 2015, IJCAI.

[9]  Albert R. Meyer,et al.  The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space , 1972, SWAT.

[10]  Bart Selman,et al.  Satisfiability Solvers , 2008, Handbook of Knowledge Representation.

[11]  Christos H. Papadimitriou,et al.  Approximability and completeness in the polynomial hierarchy , 2000 .

[12]  Stefan Szeider,et al.  Fixed-Parameter Tractable Reductions to SAT , 2014, SAT.

[13]  Stefan Szeider,et al.  Backdoors to Normality for Disjunctive Logic Programs , 2013, AAAI.

[14]  Georg Gottlob.  On minimal constraint networks , 2012, Artif. Intell..

[15]  Lane A. Hemaspaandra,et al.  SIGACT news complexity theory comun 37 , 2002, SIGA.

[16]  Georg Gottlob,et al.  Fixed-Parameter Complexity in AI and Nonmonotonic Reasoning , 1999, LPNMR.

[17]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[18]  Rolf Niedermeier,et al.  Invitation to Fixed-Parameter Algorithms , 2006 .

[19]  Celia Wrathall,et al.  Complete Sets and the Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[20]  G. S. Tseitin.  On the Complexity of Derivation in Propositional Calculus , 1983 .

[21]  Jörg Flum,et al.  Describing parameterized complexity classes , 2003, Inf. Comput..

[22]  Michael Sipser,et al.  Introduction to the Theory of Computation , 1996, SIGA.

[23]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[24]  Victor W. Marek,et al.  Stable models and an alternative logic programming paradigm , 1998, The Logic Programming Paradigm.

[25]  Carsten Sinz,et al.  Towards an Optimal CNF Encoding of Boolean Cardinality Constraints , 2005, CP.

[26]  Stefan Szeider,et al.  Machine Characterizations for Parameterized Complexity Classes Beyond Para-NP , 2015, SOFSEM.

[27]  Sloman,et al.  Automation of Reasoning , 1983, Symbolic Computation.

[28]  Armin Biere,et al.  Symbolic Model Checking without BDDs , 1999, TACAS.

[29]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[30]  Stefan Szeider,et al.  Parameterized Complexity Results for Agenda Safety in Judgment Aggregation , 2015, AAMAS.

[31]  Inês Lynce,et al.  Conflict-Driven Clause Learning SAT Solvers , 2009, Handbook of Satisfiability.

[32]  Klaus Nehring,et al.  The structure of strategy-proof social choice - Part I: General characterization and possibility results on median spaces , 2007, J. Econ. Theory.

[33]  Ronald de Haan.  An Overview of Non-Uniform Parameterized Complexity , 2015, Electron. Colloquium Comput. Complex..

[34]  Russell Impagliazzo,et al.  Which Problems Have Strongly Exponential Complexity? , 2001, J. Comput. Syst. Sci..

[35]  M. Schaefer,et al.  Completeness in the Polynomial-Time Hierarchy A Compendium ∗ , 2008 .

[36]  Stefan Szeider,et al.  The Parameterized Complexity of Reasoning Problems Beyond NP , 2013, KR.

[37]  Armin Biere,et al.  Bounded model checking , 2003, Adv. Comput..

[38]  Ulrich Endriss,et al.  Complexity of Judgment Aggregation , 2012, J. Artif. Intell. Res..

[39]  Georg Gottlob,et al.  The Complexity of Logic-Based Abduction , 1993, STACS.

[40]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[41]  Joao Marques-Silva,et al.  Anatomy and Empirical Evaluation of Modern SAT Solvers , 2011, Bull. EATCS.

[42]  Stefan Szeider,et al.  Parameterized Complexity Results for Symbolic Model Checking of Temporal Logics , 2016, KR.

[43]  Stefan Rümmele,et al.  Backdoors to Abduction , 2013, IJCAI.

[44]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[45]  Toby Walsh,et al.  Handbook of Satisfiability: Volume 185 Frontiers in Artificial Intelligence and Applications , 2009 .

[46]  Joël Ouaknine,et al.  Linear Completeness Thresholds for Bounded Model Checking , 2011, CAV.

[47]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[48]  Stefan Szeider,et al.  Compendium of Parameterized Problems at Higher Levels of the Polynomial Hierarchy , 2019, Electron. Colloquium Comput. Complex..

[49]  Sharad Malik,et al.  Boolean satisfiability from theoretical hardness to practical success , 2009, Commun. ACM.

[50]  Oded Goldreich,et al.  Computational complexity - a conceptual perspective , 2008 .

[51]  Miroslaw Truszczynski,et al.  Answer set programming at a glance , 2011, Commun. ACM.

[52]  Ronald Fagin,et al.  The Closure of Monadic NP , 2000, J. Comput. Syst. Sci..