Circuit Complexity of Bounded Planar Cutwidth Graph Matching

TL;DR

This paper proves that the perfect matching problem in bounded planar cutwidth bipartite graphs cannot be solved in AC$^0$ with prime power moduli, establishing near-tight bounds and implications for circuit complexity classes.

Contribution

It disproves the conjecture that the problem is in AC$^0$, showing it is not in AC$^0[p^{eta}]$ for any prime p, and provides new lower bounds using reductions from Parity.

Findings

The problem is not in AC$^0[p^{eta}]$ for any prime p.

Provides a reduction from Parity to BGGM.

Establishes a separation of AC$^0[m]$ from P.

Abstract

Recently, perfect matching in bounded planar cutwidth bipartite graphs (\BGGM) was shown to be in ACC$^0$ by Hansen et al.. They also conjectured that the problem is in AC$^0$. In this paper, we disprove their conjecture by showing that the problem is not in AC$^0[p^{\alpha}]$ for every prime $p$. Our results show that the previous upper bound is almost tight. Our techniques involve giving a reduction from Parity to BGGM. A further improvement in lower bounds is difficult since we do not have an algebraic characterization for AC$^0[m]$ where $m$ is not a prime power. Moreover, this will also imply a separation of AC$^0[m]$ from P. Our results also imply a better lower bound for perfect matching in general bounded planar cutwidth graphs.