Circuit Complexity of Bounded Planar Cutwidth Graph Matching

Recently, perfect matching in bounded planar cutwidth bipartite graphs (BGGM) was shown to be in ACC by Hansen et al. [8]. They also conjectured that the problem is in AC. In this paper, we disprove their conjecture by showing that the problem is not in AC[p] 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[m] where m is not a prime power. Moreover, this will also imply a separation of AC[m] from P. Our results also imply a better lower bound for perfect matching in general bounded planar cutwidth graphs.