Valiant's theory of holographic algorithms is a novel methodology
to achieve exponential speed-ups in computation. A fundamental
parameter in holographic algorithms is the dimension of the linear basis
vectors.
We completely resolve the problem of the power of higher dimensional
bases. We prove that 2-dimensional bases are universal for
holographic ... more >>>

We give a classification of block-wise symmetric signatures
in the theory of matchgate computations. The main proof technique
is matchgate identities, a.k.a. useful Grassmann-Pl\"{u}cker
identities.

Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This ... more >>>

We develop the theory of holographic algorithms. We give
characterizations of algebraic varieties of realizable
symmetric generators and recognizers on the basis manifold,
and a polynomial time decision algorithm for the
simultaneous realizability problem.
Using the general machinery we are able to give
unexpected holographic algorithms for
some counting problems, ... more >>>

The most intriguing aspect of the new theory of matchgate computations and holographic algorithms by Valiant~\cite{Valiant:Quantum} \cite{Valiant:Holographic} is that its reach and ultimate capability are wide open. The methodology produces unexpected polynomial time algorithms solving problems which seem to require exponential time. To sustain our belief in P $\not =$ ... more >>>

We establish a 1-1 correspondence between Valiant's
character theory of matchgate/matchcircuit
and his signature theory of planar-matchgate/matchgrid,
thus unifying the two theories in expressibility.
Previously we had established a complete characterization
of general matchgates, in terms of a set of
useful Grassmann-Pl{\"u}cker identities.
With this correspondence,
we give a corresponding ... more >>>

Valiant has proposed a new theory of algorithmic
computation based on perfect matchings and the Pfaffian.
We study the properties of {\it matchgates}---the basic
building blocks in this new theory. We give a set of
algebraic identities
which completely characterize these objects in terms of
the ... more >>>

We propose matchgate tensors as a natural and proper language
to develop Valiant's new theory of Holographic Algorithms.
We give a treatment of the central theorem in this theory---the Holant
Theorem---in terms of matchgate tensors.
Some generalizations are presented.

We show that the class ${\rm S}_2^p$ is a subclass of
${{\rm ZPP}^{\rm NP}}$. The proof uses universal hashing, approximate counting
and witness sampling. As a consequence, a collapse first noticed by
Samik Sengupta that the assumption NP has small circuits collapses
PH to ${\rm S}_2^p$
becomes the strongest version ... more >>>

We generalize the construction of Gabber and Galil
to essentially every unimodular matrix in $SL_2(\Z)$. It is shown that
every parabolic
or hyperbolic fractional linear transformation explicitly
defines an expander of bounded degree
and constant expansion. Thus all but a vanishingly small fraction
of unimodular matrices define expanders.

We study the complexity of the circuit minimization problem:
given the truth table of a Boolean function f and a parameter s, decide
whether f can be realized by a Boolean circuit of size at most s. We argue
why this problem is unlikely to be in P (or ... more >>>

We survey some recent developments in the study of
the complexity of lattice problems. After a discussion of some
problems on lattices which can be algorithmically solved
efficiently, our main focus is the recent progress on complexity
results of intractability. We will discuss Ajtai's worst-case/
average-case connections, NP-hardness and non-NP-hardness,
more >>>

We prove a new transference theorem in the geometry of numbers,
giving optimal bounds relating the successive minima of a lattice
with the minimal length of generating vectors of its dual.
It generalizes the transference theorem due to Banaszczyk.
We also prove a stronger bound for the special class of ... more >>>

Recently Ajtai showed that
to approximate the shortest lattice vector in the $l_2$-norm within a
factor $(1+2^{-\mbox{\tiny dim}^k})$, for a sufficiently large
constant $k$, is NP-hard under randomized reductions.
We improve this result to show that
to approximate a shortest lattice vector within a
factor $(1+ \mbox{dim}^{-\epsilon})$, for any
$\epsilon>0$, ... more >>>

This paper proves that if strong pseudorandom number generators or
one-way functions exist, then the class of languages that have
polynomial-sized circuits is not small within exponential
time, in terms of the resource-bounded measure theory of Lutz.
More precisely, if for some \epsilon > 0 there exist nonuniformly
2^{n^{\epsilon}}-hard PSRGs, ... more >>>

The modular group occupies a central position in many branches of
mathematical sciences. In this paper we give average polynomial-time
algorithms for the unbounded and bounded membership problems for
finitely generated subgroups of the modular group. The latter result
affirms a conjecture of Gurevich.