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 weighted 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]\( \,=\,\)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[t] \(\subseteq \) W[SAT] for every \(t \in \mathbb {N}\).
Supported by CAPES, CNPq and FAPERJ.
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A gate is called large if its fan-in exceeds some bound, which is typically considered to be two.
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Paranhos, R.M., Nascimento Silva, J.C., Souza, U.S., Ochi, L.S. (2021). Parameterized Complexity Classes Defined by Threshold Circuits: Using Sorting Networks to Show Collapses with W-hierarchy Classes. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Combinatorial Optimization and Applications. COCOA 2021. Lecture Notes in Computer Science(), vol 13135. Springer, Cham. https://doi.org/10.1007/978-3-030-92681-6_28
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DOI: https://doi.org/10.1007/978-3-030-92681-6_28
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