102 research outputs found
Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree
In this paper we study the problem of deterministic factorization of sparse
polynomials. We show that if is a
polynomial with monomials, with individual degrees of its variables bounded
by , then can be deterministically factored in time . Prior to our work, the only efficient factoring algorithms known for
this class of polynomials were randomized, and other than for the cases of
and , only exponential time deterministic factoring algorithms were
known.
A crucial ingredient in our proof is a quasi-polynomial sparsity bound for
factors of sparse polynomials of bounded individual degree. In particular we
show if is an -sparse polynomial in variables, with individual
degrees of its variables bounded by , then the sparsity of each factor of
is bounded by . This is the first nontrivial bound on
factor sparsity for . Our sparsity bound uses techniques from convex
geometry, such as the theory of Newton polytopes and an approximate version of
the classical Carath\'eodory's Theorem.
Our work addresses and partially answers a question of von zur Gathen and
Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the
sparsity of factors of sparse polynomials
A Matrix Hyperbolic Cosine Algorithm and Applications
In this paper, we generalize Spencer's hyperbolic cosine algorithm to the
matrix-valued setting. We apply the proposed algorithm to several problems by
analyzing its computational efficiency under two special cases of matrices; one
in which the matrices have a group structure and an other in which they have
rank-one. As an application of the former case, we present a deterministic
algorithm that, given the multiplication table of a finite group of size ,
it constructs an expanding Cayley graph of logarithmic degree in near-optimal
O(n^2 log^3 n) time. For the latter case, we present a fast deterministic
algorithm for spectral sparsification of positive semi-definite matrices, which
implies an improved deterministic algorithm for spectral graph sparsification
of dense graphs. In addition, we give an elementary connection between spectral
sparsification of positive semi-definite matrices and element-wise matrix
sparsification. As a consequence, we obtain improved element-wise
sparsification algorithms for diagonally dominant-like matrices.Comment: 16 pages, simplified proof and corrected acknowledging of prior work
in (current) Section
Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance
Structural balance theory studies stability in networks. Given a -vertex
complete graph whose edges are labeled positive or negative, the
graph is considered \emph{balanced} if every triangle either consists of three
positive edges (three mutual ``friends''), or one positive edge and two
negative edges (two ``friends'' with a common ``enemy''). From a computational
perspective, structural balance turns out to be a special case of correlation
clustering with the number of clusters at most two. The two main algorithmic
problems of interest are: detecting whether a given graph is balanced, or
finding a partition that approximates the \emph{frustration index},
i.e., the minimum number of edge flips that turn the graph balanced.
We study these problems in the streaming model where edges are given one by
one and focus on \emph{memory efficiency}. We provide randomized single-pass
algorithms for: determining whether an input graph is balanced with
memory, and finding a partition that induces a -approximation to the frustration index with memory. We further provide several new lower bounds,
complementing different aspects of our algorithms such as the need for
randomization or approximation.
To obtain our main results, we develop a method using pseudorandom generators
(PRGs) to sample edges between independently-chosen \emph{vertices} in graph
streaming. Furthermore, our algorithm that approximates the frustration index
improves the running time of the state-of-the-art correlation clustering with
two clusters (Giotis-Guruswami algorithm [SODA 2006]) from
to time for
-approximation. These results may be of independent interest
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