Complexity Classes and Theories for the Comparator Circuit Value Problem

TL;DR

This paper explores the complexity class CC related to comparator circuit problems, offering alternative definitions, a formal logical theory, and simplified proofs for problem completeness within this class.

Contribution

It introduces alternative definitions of CC, develops a two-sorted logical theory VCC*, and simplifies proofs of problem completeness in the class.

Findings

Defined alternative reducibility-based versions of CC

Developed a formal two-sorted theory VCC*

Simplified proofs of CC-completeness for stable marriage and maximal matching

Abstract

Subramanian defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem. He proved that several other problems are complete for CC, including the stable marriage problem, and finding the lexicographical first maximal matching in a bipartite graph. We suggest alternative definitions of CC based on different reducibilities and introduce a two-sorted theory VCC* based on one of them. We sharpen and simplify Subramanian's completeness proofs for the above two problems and formalize them in VCC*.