3,186 research outputs found
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation
We present an efficient proof system for Multipoint Arithmetic Circuit
Evaluation: for every arithmetic circuit of size and
degree over a field , and any inputs ,
the Prover sends the Verifier the values and a proof of length, and
the Verifier tosses coins and can check the proof in about time, with probability of error less than .
For small degree , this "Merlin-Arthur" proof system (a.k.a. MA-proof
system) runs in nearly-linear time, and has many applications. For example, we
obtain MA-proof systems that run in time (for various ) for the
Permanent, Circuit-SAT for all sublinear-depth circuits, counting
Hamiltonian cycles, and infeasibility of - linear programs. In general,
the value of any polynomial in Valiant's class can be certified
faster than "exhaustive summation" over all possible assignments. These results
strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed
by Russell Impagliazzo and others.
We also give a three-round (AMA) proof system for quantified Boolean formulas
running in time, nearly-linear time MA-proof systems for
counting orthogonal vectors in a collection and finding Closest Pairs in the
Hamming metric, and a MA-proof system running in -time for
counting -cliques in graphs.
We point to some potential future directions for refuting the
Nondeterministic Strong ETH.Comment: 17 page
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
201
Chunky and Equal-Spaced Polynomial Multiplication
Finding the product of two polynomials is an essential and basic problem in
computer algebra. While most previous results have focused on the worst-case
complexity, we instead employ the technique of adaptive analysis to give an
improvement in many "easy" cases. We present two adaptive measures and methods
for polynomial multiplication, and also show how to effectively combine them to
gain both advantages. One useful feature of these algorithms is that they
essentially provide a gradient between existing "sparse" and "dense" methods.
We prove that these approaches provide significant improvements in many cases
but in the worst case are still comparable to the fastest existing algorithms.Comment: 23 Pages, pdflatex, accepted to Journal of Symbolic Computation (JSC
The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent
Polynomial identity testing and arithmetic circuit lower bounds are two
central questions in algebraic complexity theory. It is an intriguing fact that
these questions are actually related. One of the authors of the present paper
has recently proposed a "real {\tau}-conjecture" which is inspired by this
connection. The real {\tau}-conjecture states that the number of real roots of
a sum of products of sparse univariate polynomials should be polynomially
bounded. It implies a superpolynomial lower bound on the size of arithmetic
circuits computing the permanent polynomial. In this paper we show that the
real {\tau}-conjecture holds true for a restricted class of sums of products of
sparse polynomials. This result yields lower bounds for a restricted class of
depth-4 circuits: we show that polynomial size circuits from this class cannot
compute the permanent, and we also give a deterministic polynomial identity
testing algorithm for the same class of circuits.Comment: 16 page
Fast Arithmetics in Artin-Schreier Towers over Finite Fields
An Artin-Schreier tower over the finite field F_p is a tower of field
extensions generated by polynomials of the form X^p - X - a. Following Cantor
and Couveignes, we give algorithms with quasi-linear time complexity for
arithmetic operations in such towers. As an application, we present an
implementation of Couveignes' algorithm for computing isogenies between
elliptic curves using the p-torsion.Comment: 28 pages, 4 figures, 3 tables, uses mathdots.sty, yjsco.sty Submitted
to J. Symb. Compu
Shallow Circuits with High-Powered Inputs
A polynomial identity testing algorithm must determine whether an input
polynomial (given for instance by an arithmetic circuit) is identically equal
to 0. In this paper, we show that a deterministic black-box identity testing
algorithm for (high-degree) univariate polynomials would imply a lower bound on
the arithmetic complexity of the permanent. The lower bounds that are known to
follow from derandomization of (low-degree) multivariate identity testing are
weaker. To obtain our lower bound it would be sufficient to derandomize
identity testing for polynomials of a very specific norm: sums of products of
sparse polynomials with sparse coefficients. This observation leads to new
versions of the Shub-Smale tau-conjecture on integer roots of univariate
polynomials. In particular, we show that a lower bound for the permanent would
follow if one could give a good enough bound on the number of real roots of
sums of products of sparse polynomials (Descartes' rule of signs gives such a
bound for sparse polynomials and products thereof). In this third version of
our paper we show that the same lower bound would follow even if one could only
prove a slightly superpolynomial upper bound on the number of real roots. This
is a consequence of a new result on reduction to depth 4 for arithmetic
circuits which we establish in a companion paper. We also show that an even
weaker bound on the number of real roots would suffice to obtain a lower bound
on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs
This paper presents a systematic methodology based on the algebraic theory of
signal processing to classify and derive fast algorithms for linear transforms.
Instead of manipulating the entries of transform matrices, our approach derives
the algorithms by stepwise decomposition of the associated signal models, or
polynomial algebras. This decomposition is based on two generic methods or
algebraic principles that generalize the well-known Cooley-Tukey FFT and make
the algorithms' derivations concise and transparent. Application to the 16
discrete cosine and sine transforms yields a large class of fast algorithms,
many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar
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