Parameterised Counting in Logspace

Abstract

Logarithmic space-bounded complexity classes such as textbf{L} and textbf{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. They defined the operators textbf{para}_{textbf{W}} and textbf{para}_beta 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 textbf{para}_{textbf{W}} and textbf{para}_beta by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, textbf{para}_{{textbf{W}}[1]} and textbf{para}_{beta {textbf{tail}}}. Then, we consider counting versions of all four operators and apply them to the class textbf{L} . We 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 #textbf{para}_{beta {textbf{tail}}}textbf{L} -hard and can be written as the difference of two functions in #textbf{para}_{beta {textbf{tail}}}textbf{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 #textbf{para}_{beta {textbf{tail}}}textbf{L} under parameterised logspace parsimonious reductions coincides with #textbf{para}_beta textbf{L} . In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. 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. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research.