An Overview of Non-Uniform Parameterized Complexity

We consider several non-uniform variants of parameterized complexity classes that have been considered in the literature. We do so in a homogenous notation, allowing a clear comparison of the various variants. Additionally, we consider some novel (non-uniform) parameterized complexity classes that come up in the framework of parameterized knowledge compilation. We provide some (inclusion and separation) results relating the different non-uniform classes to each other. Moreover, we illustrate how these non-uniform parameterized complexity classes are useful in the setting of parameterized knowledge compilation.

[1]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[2]  Stephen R. Mahaney.  Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

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

[4]  J. Hartmanis,et al.  On the Computational Complexity of Algorithms , 1965 .

[5]  Hubie Chen.  Parameterized Compilability , 2005, IJCAI.

[6]  Eric Allender,et al.  P-Printable Sets , 1988, SIAM J. Comput..

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

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

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

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

[11]  Richard J. Lipton,et al.  Some connections between nonuniform and uniform complexity classes , 1980, STOC '80.

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

[13]  Hubie Chen,et al.  Parameter Compilation , 2015, IPEC.

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

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

[16]  Francesco M. Donini,et al.  Preprocessing of Intractable Problems , 2002, Inf. Comput..

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

[18]  Moritz Müller,et al.  Parameterized Random Complexity , 2013, Theory of Computing Systems.

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

[20]  Liming Cai,et al.  Advice Classes of Parameterized Tractability , 1997, Ann. Pure Appl. Log..

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

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

[23]  Jiri Becvar,et al.  Review: J. Hartmanis, R. E. Stearns, On the Computational Complexity of Algorithms , 1967 .

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

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

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

[27]  Michael R. Fellows,et al.  On the parameterized complexity of multiple-interval graph problems , 2009, Theor. Comput. Sci..

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