3,885 research outputs found
An average-case depth hierarchy theorem for Boolean circuits
We prove an average-case depth hierarchy theorem for Boolean circuits over
the standard basis of , , and gates.
Our hierarchy theorem says that for every , there is an explicit
-variable Boolean function , computed by a linear-size depth- formula,
which is such that any depth- circuit that agrees with on fraction of all inputs must have size This
answers an open question posed by H{\aa}stad in his Ph.D. thesis.
Our average-case depth hierarchy theorem implies that the polynomial
hierarchy is infinite relative to a random oracle with probability 1,
confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result
to show that there is no "approximate converse" to the results of Linial,
Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus
answering a question posed by O'Donnell, Kalai, and Hatami.
A key ingredient in our proof is a notion of \emph{random projections} which
generalize random restrictions
Average-case complexity of detecting cliques
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 79-83).The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for k-CLIQUE, as this problem is known. Our results show that, in certain models of computation, solving k-CLIQUE in the average case requires Q(nk/4) resources (moreover, k/4 is tight). Here the models of computation are bounded-depth Boolean circuits and unbounded-depth monotone circuits, the complexity measure is the number of gates, and the input distributions are random graphs with an appropriate density of edges. Such random graphs (the well-studied Erdos-Renyi random graphs) are widely believed to be a source of computationally hard instances for clique problems (as Karp suggested in 1976). Our results are the first unconditional lower bounds supporting this hypothesis. For bounded-depth Boolean circuits, our average-case hardness result significantly improves the previous worst-case lower bounds of Q(nk/Poly(d)) for depth-d circuits. In particular, our lower bound of Q(nk/ 4 ) has no noticeable dependence on d for circuits of depth d ; k- log n/log log n, thus bypassing the previous "size-depth tradeoffs". As a consequence, we obtain a novel Size Hierarchy Theorem for uniform AC0 . A related application answers a longstanding open question in finite model theory (raised by Immerman in 1982): we show that the hierarchy of bounded-variable fragments of first-order logic is strict on finite ordered graphs. Additional results of this thesis characterize the average-case descriptive complexity of k-CLIQUE through the lens of first-order logic.by Benjamin Rossman.Ph.D
A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits
We study the size blow-up that is necessary to convert an algebraic circuit
of product-depth to one of product-depth in the multilinear
setting.
We show that for every positive
there is an explicit multilinear polynomial on variables
that can be computed by a multilinear formula of product-depth and
size , but not by any multilinear circuit of product-depth and
size less than . This result is tight up to the
constant implicit in the double exponent for all
This strengthens a result of Raz and Yehudayoff (Computational Complexity
2009) who prove a quasipolynomial separation for constant-depth multilinear
circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an
exponential separation in the case
Our separating examples may be viewed as algebraic analogues of variants of
the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan
(STOC 2016), who used them to prove lower bounds for constant-depth Boolean
circuits
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Near-optimal small-depth lower bounds for small distance connectivity
We show that any depth- circuit for determining whether an -node graph
has an -to- path of length at most must have size
. The previous best circuit size lower bounds for this
problem were (due to Beame, Impagliazzo, and Pitassi
[BIP98]) and (following from a recent formula size
lower bound of Rossman [Ros14]). Our lower bound is quite close to optimal,
since a simple construction gives depth- circuits of size
for this problem (and strengthening our bound even to
would require proving that undirected connectivity is not in )
Our proof is by reduction to a new lower bound on the size of small-depth
circuits computing a skewed variant of the "Sipser functions" that have played
an important role in classical circuit lower bounds [Sip83, Yao85, H{\aa}s86].
A key ingredient in our proof of the required lower bound for these Sipser-like
functions is the use of \emph{random projections}, an extension of random
restrictions which were recently employed in [RST15]. Random projections allow
us to obtain sharper quantitative bounds while employing simpler arguments,
both conceptually and technically, than in the previous works [Ajt89, BPU92,
BIP98, Ros14]
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
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