Parameterised Counting in Logspace

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 paraW and paraBeta 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 paraW and paraBeta by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, paraW[1] and paraBeta-Tail. 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 #paraBeta-Tail-L-hard and can be written as the difference of two functions in #paraBetaTail-L. For example, we show that the closure of #paraBetaTail-L under parameterised logspace parsimonious reductions coincides with #paraBeta-L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism. 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 underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research.