Circuit Complexity of Bounded Planar Cutwidth Graph Matching
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Recently, perfect matching in bounded planar cutwidth bipartite graphs () 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^α] 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.
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