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On the constant-depth complexity of k-clique

Published: 17 May 2008 Publication History

Abstract

We prove a lower bound of ω(nk/4) on the size of constant-depth circuits solving the k-clique problem on n-vertex graphs (for every constant k). This improves a lower bound of ω(nk/89d2) due to Beame where d is the circuit depth. Our lower bound has the advantage that it does not depend on the constant d in the exponent of n, thus breaking the mold of the traditional size-depth tradeoff.
Our k-clique lower bound derives from a stronger result of independent interest. Suppose fn :0,1n/2 → {0,1} is a sequence of functions computed by constant-depth circuits of size O(nt). Let G be an Erdos-Renyi random graph with vertex set {1,...,n} and independent edge probabilities n where α ≤ 1/2t-1. Let A be a uniform random k-element subset of {1,...,n} (where k is any constant independent of n) and let KA denote the clique supported on A. We prove that fn(G) = fn(G ∪ KA) asymptotically almost surely.
These results resolve a long-standing open question in finite model theory (going back at least to Immerman in 1982). The m-variable fragment of first-order logic, denoted by FOm, consists of the first-order sentences which involve at most m variables. Our results imply that the bounded variable hierarchy FO1 ⊂ FO2 ⊂ ... ⊂ FOm ⊂ ... is strict in terms of expressive power on finite ordered graphs. It was previously unknown that FO3 is less expressive than full first-order logic on finite ordered graphs.

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cover image ACM Conferences
STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing
May 2008
712 pages
ISBN:9781605580470
DOI:10.1145/1374376
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Published: 17 May 2008

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  1. ac0
  2. bounded variable hierarchy
  3. circuit complexity
  4. constant-depth circuits
  5. first-order logic
  6. k-clique

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STOC '08: Symposium on Theory of Computing
May 17 - 20, 2008
British Columbia, Victoria, Canada

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  • (2023)Constant-Depth Circuits vs. Monotone CircuitsProceedings of the conference on Proceedings of the 38th Computational Complexity Conference10.4230/LIPIcs.CCC.2023.29(1-37)Online publication date: 17-Jul-2023
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