32,213 research outputs found
Progress on Polynomial Identity Testing - II
We survey the area of algebraic complexity theory; with the focus being on
the problem of polynomial identity testing (PIT). We discuss the key ideas that
have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve
Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs
A read-once oblivious arithmetic branching program (ROABP) is an arithmetic
branching program (ABP) where each variable occurs in at most one layer. We
give the first polynomial time whitebox identity test for a polynomial computed
by a sum of constantly many ROABPs. We also give a corresponding blackbox
algorithm with quasi-polynomial time complexity . In both the
cases, our time complexity is double exponential in the number of ROABPs.
ROABPs are a generalization of set-multilinear depth- circuits. The prior
results for the sum of constantly many set-multilinear depth- circuits were
only slightly better than brute-force, i.e. exponential-time.
Our techniques are a new interplay of three concepts for ROABP: low
evaluation dimension, basis isolating weight assignment and low-support rank
concentration. We relate basis isolation to rank concentration and extend it to
a sum of two ROABPs using evaluation dimension (or partial derivatives).Comment: 22 pages, Computational Complexity Conference, 201
Quantum Query Complexity of Multilinear Identity Testing
Motivated by the quantum algorithm in \cite{MN05} for testing commutativity
of black-box groups, we study the following problem: Given a black-box finite
ring where is an additive
generating set for and a multilinear polynomial over
also accessed as a black-box function (where we allow the
indeterminates to be commuting or noncommuting), we study the
problem of testing if is an \emph{identity} for the ring . More
precisely, the problem is to test if for all .
We give a quantum algorithm with query complexity assuming . Towards a lower bound,
we also discuss a reduction from a version of -collision to this problem.
We also observe a randomized test with query complexity and constant
success probability and a deterministic test with query complexity.Comment: 12 page
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
Algorithms for group isomorphism via group extensions and cohomology
The isomorphism problem for finite groups of order n (GpI) has long been
known to be solvable in time, but only recently were
polynomial-time algorithms designed for several interesting group classes.
Inspired by recent progress, we revisit the strategy for GpI via the extension
theory of groups.
The extension theory describes how a normal subgroup N is related to G/N via
G, and this naturally leads to a divide-and-conquer strategy that splits GpI
into two subproblems: one regarding group actions on other groups, and one
regarding group cohomology. When the normal subgroup N is abelian, this
strategy is well-known. Our first contribution is to extend this strategy to
handle the case when N is not necessarily abelian. This allows us to provide a
unified explanation of all recent polynomial-time algorithms for special group
classes.
Guided by this strategy, to make further progress on GpI, we consider
central-radical groups, proposed in Babai et al. (SODA 2011): the class of
groups such that G mod its center has no abelian normal subgroups. This class
is a natural extension of the group class considered by Babai et al. (ICALP
2012), namely those groups with no abelian normal subgroups. Following the
above strategy, we solve GpI in time for central-radical
groups, and in polynomial time for several prominent subclasses of
central-radical groups. We also solve GpI in time for
groups whose solvable normal subgroups are elementary abelian but not
necessarily central. As far as we are aware, this is the first time there have
been worst-case guarantees on a -time algorithm that tackles
both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation,
with some new result
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