The Complexity of the Comparator Circuit Value Problem

TL;DR

This paper explores the complexity class CC, defined via comparator circuits, providing multiple characterizations, evidence of its incomparability with NC, and connections to problems like stable marriage.

Contribution

It introduces alternative definitions and models for CC, demonstrates its robustness, and provides evidence for its conjectured incomparability with NC through oracle constructions.

Findings

CC is a robust complexity class with multiple equivalent definitions.

Relativized worlds show CC and NC are incomparable.

Simplified proofs and connections to stable marriage algorithms are provided.

Abstract

In 1990 Subramanian defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem (CCV). He and Mayr showed that NL \subseteq CC \subseteq P, and proved that in addition to CCV several other problems are complete for CC, including the stable marriage problem, and finding the lexicographically first maximal matching in a bipartite graph. We are interested in CC because we conjecture that it is incomparable with the parallel class NC which also satisfies NL \subseteq NC \subseteq P, and note that this conjecture implies that none of the CC-complete problems has an efficient polylog time parallel algorithm. We provide evidence for our conjecture by giving oracle settings in which relativized CC and relativized NC are incomparable. We give several alternative definitions of CC, including (among others) the class of problems computed by uniform polynomial-size families of comparator circuits supplied with copies of the input and its negation, the class of problems AC^0-reducible to CCV, and the class of problems computed by uniform AC^0 circuits with CCV gates. We also give a machine model for CC, which corresponds to its characterization as log-space uniform polynomial-size families of comparator circuits. These various characterizations show that CC is a robust class. The main technical tool we employ is universal comparator circuits. Other results include a simpler proof of NL \subseteq CC, and an explanation of the relation between the Gale-Shapley algorithm and Subramanian's algorithm for stable marriage. This paper continues the previous work of Cook, L\^e and Ye which focused on Cook-Nguyen style uniform proof complexity, answering several open questions raised in that paper.