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

@article{Chen2016NearoptimalSL,
  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},
  year={2016}
}
  • 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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References

SHOWING 1-10 OF 21 REFERENCES

Formulas vs. circuits for small distance connectivity

TLDR
Half of the proof shows that bounded-depth formulas solving Distance k(n) Connectivity imply upper bounds on pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size n via the operations of union and relational join, subject to certain density constraints.

Improved depth lower bounds for small distance connectivity

TLDR
A new form of “switching lemma” which combines some of the features of iteratively shortening terms due to Furst et al. (1984), Yao (1985), Håstad (1987), and Cai (1986) that have been the methods of choice for subsequent results is used.

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

TLDR
The average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Hastad [Has86a], Cai [Cai86], and Babai [Bab87].

Size-Depth Tradeoffs for Threshold Circuits

TLDR
The lower bound implies an affirmative answer to the conjecture of Paturi and Saks that a bounded-depth threshold circuit that computes parity requires a superlinear number of edges and is the first superlinear lower bound for an explicit function that holds for any fixed depth and the first that applies to threshold circuits with unrestricted weights.

Approximation and Small-Depth Frege Proofs

TLDR
This paper demonstrates how to eliminate the nonstandard model theory, including the nonconstructive use of the compactness theorem, from Ajtai's lower bound by introducing the notion of an “approximate proof.

Computational limitations of small-depth circuits

TLDR
The techniques described in "Computational Limitations for Small Depth Circuits" can be used to demonstrate almost optimal lower bounds on the size of small depth circuits computing several different functions, such as parity and majority.

On monotone formulae with restricted depth

TLDR
It is shown that any function with a formula of size n (and any depth) has a &sgr;<subscrpt; k-formula of size exp o(n<supscrpt>1/(k−1)</supsCrpt) and the hierarchy theorem is the best possible.

First-Order Definability on Finite Structures

  • M. Ajtai
  • Mathematics
    Ann. Pure Appl. Log.
  • 1989

Separating the polynomial-time hierarchy by oracles

  • A. Yao
  • Computer Science, Mathematics
  • 1985
We present exponential lower bounds on the size of depth-k Boolean circuits for computing certain functions. These results imply that there exists an oracle set A such that, relative to A, all the

With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy

  • Jin-Yi Cai
  • Computer Science, Mathematics
    STOC '86
  • 1986
TLDR
It is shown that a random oracle set A separates PSPACE from the entire polynomial-time hierarchy with probability one as a consequence of how much error a fixed depth Boolean circuit must make in computing the parity function.