509 research outputs found
On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries
We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and
prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We
prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution
to a convex optimization problem. We show that if Sigma(R, C) is sufficiently
large, then a random matrix D in Sigma(R, C) sampled from the uniform
probability measure in Sigma(R,C) with high probability is close to a
particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among
all matrices with row sums R, column sums C and entries between 0 and 1.
Similar results are obtained for 0-1 matrices with prescribed row and column
sums and assigned zeros in some positions.Comment: 26 pages, proofs simplified, results strengthene
Brunn-Minkowski Inequalities for Contingency Tables and Integer Flows
Given a non-negative mxn matrix W=(w_ij) and positive integer vectors R=(r_1,
>..., r_m) and C=(c_1, ..., c_n), we consider the total weight T(R, C; W) of
mxn non-negative integer matrices (contingency tables) D with the row sums r_i,
the column sums c_j, and the weight of D=(d_ij) equal to product of w_ij^d_ij.
In particular, if W is a 0-1 matrix, T(R, C; W) is the number of integer
feasible flows in a bipartite network. We prove a version of the
Brunn-Minkowski inequality relating the numbers T(R, C; W) and T(R_k, C_k; W),
where (R, C) is a convex combination of (R_k, C_k) for k=1, ..., p.Comment: 16 page
Computing the partition function of a polynomial on the Boolean cube
For a polynomial f: {-1, 1}^n --> C, we define the partition function as the
average of e^{lambda f(x)} over all points x in {-1, 1}^n, where lambda in C is
a parameter. We present a quasi-polynomial algorithm, which, given such f,
lambda and epsilon >0 approximates the partition function within a relative
error of epsilon in N^{O(ln n -ln epsilon)} time provided |lambda| < 1/(2 L
sqrt{deg f}), where L=L(f) is a parameter bounding the Lipschitz constant of f
from above and N is the number of monomials in f. As a corollary, we obtain a
quasi-polynomial algorithm, which, given such an f with coefficients +1 and -1
and such that every variable enters not more than 4 monomials, approximates the
maximum of f on {-1, 1}^n within a factor of O(sqrt{deg f}/delta), provided the
maximum is N delta for some 0< delta <1. If every variable enters not more than
k monomials for some fixed k > 4, we are able to establish a similar result
when delta > (k-1)/k.Comment: The final version of this paper is due to be published in the
collection of papers "A Journey through Discrete Mathematics. A Tribute to
Jiri Matousek" edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, to
be published by Springe
A simple polynomial time algorithm to approximate the permanent within a simply exponential factor
We present a simple randomized polynomial time algorithm to approximate the
mixed discriminant of positive semidefinite matrices within a
factor . Consequently, the algorithm allows us to approximate in
randomized polynomial time the permanent of a given non-negative
matrix within a factor . When applied to approximating the permanent,
the algorithm turns out to be a simple modification of the well-known
Godsil-Gutman estimator
Approximations of convex bodies by polytopes and by projections of spectrahedra
We prove that for any compact set B in R^d and for any epsilon >0 there is a
finite subset X of B of |X|=d^{O(1/epsilon^2)} points such that the maximum
absolute value of any linear function ell: R^d --> R on X approximates the
maximum absolute value of ell on B within a factor of epsilon sqrt{d}. We also
discuss approximations of convex bodies by projections of spectrahedra, that
is, by projections of sections of the cone of positive semidefinite matrices by
affine subspaces.Comment: 13 pages, some improvements, acknowledgment adde
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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