Parameterized Complexity Classes Defined by Threshold Circuits and Their Connection with Sorting Networks

Abstract

The main complexity classes of the Parameterized Intractability Theory are based on weighted Boolean circuit satisfiability problems and organized into a hierarchy so-called W-hierarchy. The W-hierarchy enables fine-grained complexity analyses of parameterized problems that are unlikely to belong to the FPT class. In this paper, we introduce the Th-hierarchy, a natural generalization of the W-hierarchy defined by unweighted threshold circuit satisfiability problems. Investigating the relationship between Th-hierarchy and W-hierarchy, we discuss the complexity of transforming Threshold circuits into Boolean circuits, and observe that sorting networks are powerful tools to handle such transformations. First, we show that these hierarchies collapse at the last level (W[P][Formula: see text][Formula: see text][Formula: see text]Th[P]). After that, we present a time complexity analysis of an AKS sorting network construction, which supports some of our results. Finally, we prove that Th[[Formula: see text]] [Formula: see text] W[SAT] for every [Formula: see text]. As a by-product, our studies suggest that it is relevant to consider a new class based on logarithmic depth circuits in the W-hierarchy.