139 research outputs found
Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials
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 polynomial. We
first prove that the first problem is \#P-hard and then devise a
upper bound for this problem for any polynomial represented by an arithmetic
circuit of size . Later, this upper bound is improved to for
polynomials. We then design fully polynomial-time randomized
approximation schemes for this problem for polynomials. On the
negative side, we prove that, even for polynomials with terms of
degree , the first problem cannot be approximated at all for any
approximation factor , nor {\em "weakly approximated"} in a much relaxed
setting, unless P=NP. For the second problem, we first give a polynomial time
-approximation algorithm for polynomials with terms of
degrees no more a constant . On the inapproximability side, we
give a lower bound, for any on the
approximation factor for polynomials. When terms in these
polynomials are constrained to degrees , we prove a lower
bound, assuming ; and a higher lower bound, assuming the
Unique Games Conjecture
Monomial Testing and Applications
In this paper, we devise two algorithms for the problem of testing
-monomials of degree in any multivariate polynomial represented by a
circuit, regardless of the primality of . One is an time
randomized algorithm. The other is an time deterministic
algorithm for the same -monomial testing problem but requiring the
polynomials to be represented by tree-like circuits. Several applications of
-monomial testing are also given, including a deterministic
upper bound for the -set -packing problem.Comment: 17 pages, 4 figures, submitted FAW-AAIM 2013. arXiv admin note:
substantial text overlap with arXiv:1302.5898; and text overlap with
arXiv:1007.2675, arXiv:1007.2678, arXiv:1007.2673 by other author
Faster Deterministic Algorithms for Packing, Matching and -Dominating Set Problems
In this paper, we devise three deterministic algorithms for solving the
-set -packing, -dimensional -matching, and -dominating set
problems in time , and ,
respectively. Although recently there has been remarkable progress on
randomized solutions to those problems, our bounds make good improvements on
the best known bounds for deterministic solutions to those problems.Comment: ISAAC13 Submission. arXiv admin note: substantial text overlap with
arXiv:1303.047
The Complexity of Testing Monomials in Multivariate Polynomials
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 with two folds of objectives. One is to
understand how this problem relates to critical problems in complexity, and if
so to what extent. The other is to exploit possibilities of applying algebraic
properties of polynomials to the study of those problems. A series of results
about and polynomials are obtained in this paper,
laying a basis for further study along this line
Lower Bounds for Monotone Counting Circuits
A {+,x}-circuit counts a given multivariate polynomial f, if its values on
0-1 inputs are the same as those of f; on other inputs the circuit may output
arbitrary values. Such a circuit counts the number of monomials of f evaluated
to 1 by a given 0-1 input vector (with multiplicities given by their
coefficients). A circuit decides if it has the same 0-1 roots as f. We
first show that some multilinear polynomials can be exponentially easier to
count than to compute them, and can be exponentially easier to decide than to
count them. Then we give general lower bounds on the size of counting circuits.Comment: 20 page
Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations
In this paper we study multivariate polynomial functions in complex variables
and the corresponding associated symmetric tensor representations. The focus is
on finding conditions under which such complex polynomials/tensors always take
real values. We introduce the notion of symmetric conjugate forms and general
conjugate forms, and present characteristic conditions for such complex
polynomials to be real-valued. As applications of our results, we discuss the
relation between nonnegative polynomials and sums of squares in the context of
complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for
complex tensors are introduced, extending properties from the Hermitian
matrices. Finally, we discuss an important property for symmetric tensors,
which states that the largest absolute value of eigenvalue of a symmetric real
tensor is equal to its largest singular value; the result is known as Banach's
theorem. We show that a similar result holds in the complex case as well
Integration and Optimization of Multivariate Polynomials by Restriction onto a Random Subspace
We consider the problem of efficient integration of an n-variate polynomial
with respect to the Gaussian measure in R^n and related problems of complex
integration and optimization of a polynomial on the unit sphere. We identify a
class of n-variate polynomials f for which the integral of any positive integer
power f^p over the whole space is well-approximated by a properly scaled
integral over a random subspace of dimension O(log n). Consequently, the
maximum of f on the unit sphere is well-approximated by a properly scaled
maximum on the unit sphere in a random subspace of dimension O(log n). We
discuss connections with problems of combinatorial counting and applications to
efficient approximation of a hafnian of a positive matrix.Comment: 15 page
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