We show that
derandomizing polynomial identity testing over an arbitrary finite
field implies that NEXP does not have polynomial size boolean
circuits. In other words, for any finite field F(q) of size q,
$PIT_q\in NSUBEXP\Rightarrow NEXP\not\subseteq P/poly$, where
$PIT_q$ is the polynomial identity testing problem over F(q), and
NSUBEXP is ... more >>>

The bin packing problem is to find the minimum
number of bins of size one to pack a list of items with sizes
$a_1,\ldots , a_n$ in $(0,1]$. Using uniform sampling, which selects
a random element from the input list each time, we develop a
randomized $O({n(\log n)(\log\log n)\over ... more >>>

We investigate the complexity of integration and
derivative for multivariate polynomials in the standard computation
model. The integration is in the unit cube $[0,1]^d$ for a
multivariate polynomial, which has format
$f(x_1,\cdots,
x_d)=p_1(x_1,\cdots, x_d)p_2(x_1,\cdots, x_d)\cdots p_k(x_1,\cdots,
x_d)$,
where each $p_i(x_1,\cdots, x_d)=\sum_{j=1}^d q_j(x_j)$ with
all single variable polynomials $q_j(x_j)$ ... more >>>

A long standing open problem in the computational complexity theory
is to separate NE from BPP, which is a subclass of $NP_T (NP\cap P/poly)$.
In this paper, we show that $NE\not\subseteq NP_T (NP \cap$ Nonexponentially-Dense-Class),
where Nonexponentially-Dense-Class is the class of languages A without exponential density
(for ... more >>>

This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a $\Pi\Sigma\Pi$ polynomial.
We first prove ... more >>>

This paper is our second step towards developing a theory of
testing monomials in multivariate polynomials. The central
question is to ask whether a polynomial represented by an
arithmetic circuit has some types of monomials in its sum-product
expansion. The complexity aspects of this problem and its variants
have been ... more >>>

The work in this paper is to initiate a theory of testing
monomials in multivariate polynomials. The central question is to
ask whether a polynomial represented by certain economically
compact structure has a multilinear monomial in its sum-product
expansion. The complexity aspects of this problem and its variants
are investigated ... more >>>

It is well known that every finite Abelian group $G$ can be
represented as a product of cyclic groups: $G=G_1\times
G_2\times\cdots G_t$, where each $G_i$ is a cyclic group of size
$p^j$ for some prime $p$ and integer $j\ge 1$. If $a_i$ is the
generator of the cyclic group of ... more >>>

We introduce the notion of width bounded geometric separator,
develop the techniques for its existence as well as algorithm, and
apply it to obtain a $2^{O(\sqrt{n})}$ time exact algorithm for the
disk covering problem, which seeks to determine the minimal number
of fixed size disks to cover $n$ points on ... more >>>