Improved Monotone Circuit Depth Upper Bound for Directed Graph Reachability

TL;DR

This paper advances the theoretical understanding of monotone circuit complexity for directed graph reachability, providing improved upper bounds on circuit depth with both non-constructive and constructive proofs.

Contribution

It establishes a tighter upper bound on monotone circuit depth for directed reachability, improving previous results and offering a constructive proof.

Findings

Non-constructive proof of depth (1/2+o(1))(log n)^2

Constructive proof of depth (7/8+o(1))(log n)^2

Improved upper bounds on monotone circuit depth for directed reachability

Abstract

We prove that the directed graph reachability problem (transitive closure) can be solved by monotone fan-in 2 boolean circuits of depth (1/2+o(1))(log n)^2, where n is the number of nodes. This improves the previous known upper bound (1+o(1))(log n)^2. The proof is non-constructive, but we give a constructive proof of the upper bound (7/8+o(1))(log n)^2.