Parameterised Counting in Logspace

Stockhusen and Tantau (IPEC 2013) defined the operators paraW and paraBeta for parameterised space complexity classes by allowing bounded nondeterminism with multiple read and read-once access, respectively. Using these operators, they obtained characterisations for the complexity of many parameterisations of natural problems on graphs. In this article, we study the counting versions of such operators and introduce variants based on tail-nondeterminism, paraW[1] and paraBetaTail, in the setting of parameterised logarithmic space. We examine closure properties of the new classes under the central reductions and arithmetic operations. We also identify a wide range of natural complete problems for our classes in the areas of walk counting in digraphs, first-order model-checking and graph-homomorphisms. In doing so, we also see that the closure of #paraBetaTail-L under parameterised logspace parsimonious reductions coincides with #paraBeta-L. We show that the complexity of a parameterised variant of the determinant function is #paraBetaTail-L-hard and can be written as the difference of two functions in #paraBetaTail-L for (0,1)-matrices. Finally, we characterise the new complexity classes in terms of branching programs.

[1]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[2]  Heribert Vollmer,et al.  Nondeterministic NC/sup 1/ computation , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[3]  Eric Allender,et al.  The complexity of matrix rank and feasible systems of linear equations , 1999, computational complexity.

[4]  Fan Yang,et al.  Counting of Teams in First-Order Team Logics , 2019, MFCS.

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

[6]  Hubie Chen,et al.  The Fine Classification of Conjunctive Queries and Parameterized Logarithmic Space , 2013, ACM Trans. Comput. Theory.

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

[8]  Bin Fu,et al.  Circuits over PP and PL , 2000, J. Comput. Syst. Sci..

[9]  Catherine S. Greenhill,et al.  The complexity of counting graph homomorphisms , 2000 .

[10]  Till Tantau,et al.  Completeness Results for Parameterized Space Classes , 2013, IPEC.

[11]  Radu Curticapean,et al.  Counting Matchings of Size k Is W[1]-Hard , 2013, ICALP.

[12]  Marc Roth,et al.  Parameterized Counting of Trees, Forests and Matroid Bases , 2016, CSR.

[13]  Jörg Flum,et al.  Query evaluation via tree-decompositions , 2001, JACM.

[14]  Yijia Chen,et al.  Bounded nondeterminism and alternation in parameterized complexity theory , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

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

[16]  Heribert Vollmer,et al.  Counting classes and the fine structure between NC1 and L , 2012, Theor. Comput. Sci..

[17]  Michael Elberfeld,et al.  On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness , 2015, Algorithmica.

[18]  Mark Jerrum,et al.  The parameterised complexity of counting even and odd induced subgraphs , 2014, Comb..

[19]  Radu Curticapean,et al.  The simple, little and slow things count: on parameterized counting complexity , 2015, Bull. EATCS.

[20]  Max Bannach,et al.  Fast Parallel Fixed-parameter Algorithms via Color Coding , 2015, IPEC.

[21]  Catherine McCartin.  Parameterized Counting Problems , 2002, MFCS.

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

[23]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[24]  Peter Jonsson,et al.  The complexity of counting homomorphisms seen from the other side , 2004, Theor. Comput. Sci..

[25]  Yijia Chen,et al.  Some Lower Bounds in Parameterized AC^0 , 2016, MFCS.

[26]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

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

[28]  Christoph Meinel,et al.  p-Projection Reducibility and the Complexity Classes L(nonuniform) and NL(nonuniform) , 1996, MFCS.

[29]  Heribert Vollmer,et al.  Introduction to Circuit Complexity: A Uniform Approach , 2010 .

[30]  Martin Grohe.  The complexity of homomorphism and constraint satisfaction problems seen from the other side , 2007, JACM.

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

[32]  Mitsunori Ogihara.  The PL Hierarchy Collapses , 1998, SIAM J. Comput..

[33]  Meena Mahajan,et al.  Determinant: Combinatorics, Algorithms, and Complexity , 1997, Chic. J. Theor. Comput. Sci..

[34]  V Vinay.  Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[35]  Eric Allender,et al.  On TC0, AC0, and Arithmetic Circuits , 2000, J. Comput. Syst. Sci..

[36]  B. V. Raghavendra Rao,et al.  Parameterized Analogues of Probabilistic Computation , 2014, CALDAM.

[37]  Yijia Chen,et al.  Some lower bounds in parameterized ${\rm AC}^0$ , 2016 .

[38]  Jörg Flum,et al.  The Parameterized Complexity of Counting Problems , 2004, SIAM J. Comput..

[39]  Meena Mahajan,et al.  Small-Space Analogues of Valiant's Classes , 2009, FCT.