scholarly journals Lower Bounds on Arithmetic Circuits via Partial Derivatives (Preliminary Version)

1995 ◽  
Vol 2 (43) ◽  
Author(s):  
Noam Nisan ◽  
Avi Wigderson
Keyword(s):  
Lower Bounds ◽  
Linear Span ◽  
Arithmetic Circuits ◽  
Multivariate Polynomials ◽  
Symmetric Polynomials ◽  
Partial Derivatives ◽  
Matrix Products ◽  
Preliminary Version ◽  
A New Technique ◽  
Monotone Arithmetic Circuits

In this paper we describe a new technique for obtaining lower bounds on<br />restricted classes of non-monotone arithmetic circuits. The heart of this technique is a complexity measure for multivariate polynomials, based on the linear span of their partial derivatives. We use the technique to obtain new lower bounds for computing symmetric polynomials and iterated matrix products.

Lower bounds on arithmetic circuits via partial derivatives

1996 ◽  
Vol 6 (3) ◽  
pp. 217-234 ◽  
Author(s):  
Noam Nisan ◽  
Avi Wigderson
Keyword(s):  
Lower Bounds ◽  
Arithmetic Circuits ◽  
Partial Derivatives

scholarly journals Uniform derandomization from pathetic lower bounds

2012 ◽  
Vol 370 (1971) ◽  
pp. 3512-3535 ◽  
Author(s):  
Eric Allender ◽  
V. Arvind ◽  
Rahul Santhanam ◽  
Fengming Wang
Keyword(s):  
Word Problem ◽  
Lower Bounds ◽  
Black Box ◽  
Arithmetic Circuits ◽  
List Type ◽  
Probabilistic Algorithms ◽  
Constant Depth ◽  
Subexponential Time ◽  
Polynomial Size ◽  
Threshold Circuits

The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where ‘pathetic’ lower bounds of the form n 1+ ϵ would suffice to derandomize interesting classes of probabilistic algorithms. We show the following: — If the word problem over S 5 requires constant-depth threshold circuits of size n 1+ ϵ for some ϵ >0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size). — If there are no constant-depth arithmetic circuits of size n 1+ ϵ for the problem of multiplying a sequence of n  3×3 matrices, then, for every constant d , black-box identity testing for depth- d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

scholarly journals Lower Bounds for Monotone Span Programs

1994 ◽  
Vol 1 (46) ◽  
Author(s):  
Amos Beimel
Keyword(s):  
Lower Bounds ◽  
Secret Sharing ◽  
Algebraic Model ◽  
New Technique ◽  
Secret Sharing Schemes ◽  
Branching Programs ◽  
Model Of Computation ◽  
Monotone Span Programs ◽  
Span Programs ◽  
A New Technique

The model of span programs is a linear algebraic model of computation. Lower bounds for span programs imply lower bounds for contact schemes, symmetric branching programs and for formula size. Monotone span programs correspond also to linear secret-sharing schemes. We present a new technique for proving lower bounds for monotone span programs. The main result proved here yields quadratic lower bounds for the size of monotone span programs, improving on the largest previously known bounds for explicit functions. The bound is asymptotically tight for the function corresponding to a class of 4-cliques.

Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space

2015 ◽  
Vol 160 (2) ◽  
pp. 191-208 ◽  
Author(s):  
SERGEI ARTAMOSHIN
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Negative Curvature ◽  
Connected Space ◽  
Constant Negative Curvature ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalues ◽  
First Dirichlet Eigenvalue ◽  
A New Technique ◽  
Simply Connected

AbstractWe consider domains in a simply connected space of constant negative curvature and develop a new technique that improves existing classical lower bound for Dirichlet eigenvalues obtained by H. P. McKean as well as the lower bounds recently obtained by A. Savo.

scholarly journals Smaller Cuts, Higher Lower Bounds

2021 ◽  
Vol 17 (4) ◽  
pp. 1-40
Author(s):  
Amir Abboud ◽  
Keren Censor-Hillel ◽  
Seri Khoury ◽  
Ami Paz
Keyword(s):  
Lower Bounds ◽  
Vertex Cover ◽  
Shortest Paths ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Graph Problems ◽  
All Pairs Shortest Paths ◽  
Dense Graphs ◽  
Hard Problems ◽  
A New Technique

This article proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget : a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P, which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

scholarly journals Lower Bounds for Arithmetic Circuits via the Hankel Matrix

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Nathanaël Fijalkow ◽  
Guillaume Lagarde ◽  
Pierre Ohlmann ◽  
Olivier Serre
Keyword(s):  
Lower Bounds ◽  
Hankel Matrix ◽  
Arithmetic Circuits
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