Near-optimal small-depth lower bounds for small distance connectivity

  title={Near-optimal small-depth lower bounds for small distance connectivity},
  author={Xi Chen and Igor Carboni Oliveira and Rocco A. Servedio and Li-Yang Tan},
  journal={Proceedings of the forty-eighth annual ACM symposium on Theory of Computing},
  • Xi Chen, I. Oliveira, Li-Yang Tan
  • Published 24 September 2015
  • Computer Science
  • Proceedings of the forty-eighth annual ACM symposium on Theory of Computing
We show that any depth-d circuit for determining whether an n-node graph has an s-to-t path of length at most k must have size nΩ(k1/d/d) when k(n) ≤ n1/5, and nΩ(k1/5d/d) when k(n)≤ n. The previous best circuit size lower bounds were nkexp(−O(d)) (by Beame, Impagliazzo, and Pitassi (Computational Complexity 1998)) and nΩ((logk)/d) (following from a recent formula size lower bound of Rossman (STOC 2014)). Our lower bound is quite close to optimal, as a simple construction gives depth-d circuits… 

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