Guest column: a casual tour around a circuit complexity bound

@article{Williams2011GuestCA,
  title={Guest column: a casual tour around a circuit complexity bound},
  author={Ryan Williams},
  journal={ArXiv},
  year={2011},
  volume={abs/1111.1261}
}
I will discuss the recent proof that the complexity class NEXP (nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial size. The proof will be described from the perspective of someone trying to discover it. 
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Non-uniform ACC Circuit Lower Bounds
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Parameterized Graph Modification Algorithms
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TLDR
A deterministic algorithm for counting the number of satisfying assignments of any AC0[⊕] circuit C of size s and depth d over n variables in time 2n−f(n,s,d), which beats the lower bound of 2Ω(n) due to Razborov and Smolensky for large d.
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References

SHOWING 1-10 OF 59 REFERENCES
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
A Uniform Circuit Lower Bound for the Permanent
The authors show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the
Worst-Case Upper Bounds
The chapter is a survey of ideas and techniques behind satisfiability algorithms with the currently best asymptotic upper bounds on the worst-case running time. The survey also includes related
Parity, circuits, and the polynomial-time hierarchy
TLDR
A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function and connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.
A Note on Succinct Representations of Graphs
A full derandomization of schöning's k-SAT algorithm
TLDR
A deterministic version ofchopping in 1999 presented a simple randomized algorithm for k-SAT with running time a n+o(n) for a = 2(k-1)/k is given.
ON ACC and threshold circuits
  • A. Yao
  • Computer Science
    Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science
  • 1990
TLDR
It is proved that any language in ACC can be approximately computed by two-level circuits of size 2 raised to the (log n)/sup k/ power, with a symmetric-function gate at the top and only AND gates on the first level, giving the first nontrivial upper bound on the computing power of ACC circuits.
Some connections between nonuniform and uniform complexity classes
TLDR
This work aims to understand when nonuniform upper bounds can be used to obtain uniform upper bounds, and how to relate it to more common notions.
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
TLDR
It is proved that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm.
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