Upper Bounds for Monotone Planar Circuit Value and Variants

Abstract.

The P-complete Circuit Value Problem CVP, when restricted to monotone planar circuits MPCVP, is known to be in NC³, and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we re-examine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that stratified cylindrical monotone circuits can be evaluated in LogDCFL, and arbitrary cylindrical monotone circuits can be evaluated in AC¹(LogDCFL), while monotone circuits with one-input-face planar embeddings can be evaluated in LogCFL. For monotone circuits with focused embeddings, we show an upper bound of AC¹(LogDCFL). We re-examine the NC³ algorithm for general MPCVP, and note that it is in AC¹(LogCFL) = SAC². Finally, we consider extensions beyond MPCVP. We show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. Also, special kinds of arbitrary genus circuits can also be evaluated in NC. We also show that planar non-monotone circuits with polylogarithmic negation-height can be evaluated in NC.