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Parameterised Counting in Logspace

Authors Anselm Haak , Arne Meier , Om Prakash, Raghavendra Rao B. V.



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Author Details

Anselm Haak
  • Institut für Theoretische Informatik, Leibniz Universität Hannover, Germany
Arne Meier
  • Institut für Theoretische Informatik, Leibniz Universität Hannover, Germany
Om Prakash
  • Department of Computer Science and Engineering, IIT Madras, Chennai, India
Raghavendra Rao B. V.
  • Department of Computer Science and Engineering, IIT Madras, Chennai, India

Acknowledgements

The authors thank the anonymous referees for their valuable feedback.

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Anselm Haak, Arne Meier, Om Prakash, and Raghavendra Rao B. V.. Parameterised Counting in Logspace. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 40:1-40:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.40

Abstract

Logarithmic space bounded complexity classes such as L and NL play a central role in space bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators para_W and para_β for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators para_W and para_β by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, para_{W[1]} and para_{βtail}. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #para_{βtail} L-hard and can be written as the difference of two functions in #para_{βtail} L. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #para_{βtail} L under parameterised logspace parsimonious reductions coincides with #para_β L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Also, we show that the counting classes defined can naturally be characterised by parameterised variants of classes based on branching programs in analogy to the classical counting classes. Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized Complexity
  • Counting Complexity
  • Logspace

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