6,932 research outputs found
Approximating the Permanent of a Random Matrix with Vanishing Mean
We show an algorithm for computing the permanent of a random matrix with
vanishing mean in quasi-polynomial time. Among special cases are the Gaussian,
and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we
can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time
2^{O(n^{\eps})} for any small constant \eps>0. Our algorithm counters the
intuition that the permanent is hard because of the "sign problem" - namely the
interference between entries of a matrix with different signs. A major open
question then remains whether one can provide an efficient algorithm for random
matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the
baseline assumptions of the BosonSampling paradigm
Counting magic squares in quasi-polynomial time
We present a randomized algorithm, which, given positive integers n and t and
a real number 0< epsilon <1, computes the number Sigma(n, t) of n x n
non-negative integer matrices (magic squares) with the row and column sums
equal to t within relative error epsilon. The computational complexity of the
algorithm is polynomial in 1/epsilon and quasi-polynomial in N=nt, that is, of
the order N^{log N}. A simplified version of the algorithm works in time
polynomial in 1/epsilon and N and estimates Sigma(n,t) within a factor of
N^{log N}. This simplified version has been implemented. We present results of
the implementation, state some conjectures, and discuss possible
generalizations.Comment: 30 page
Enumerating contingency tables via random permanents
Given m positive integers R=(r_i), n positive integers C=(c_j) such that sum
r_i = sum c_j =N, and mn non-negative weights W=(w_{ij}), we consider the total
weight T=T(R, C; W) of non-negative integer matrices (contingency tables)
D=(d_{ij}) with the row sums r_i, column sums c_j, and the weight of D equal to
prod w_{ij}^{d_{ij}}. We present a randomized algorithm of a polynomial in N
complexity which computes a number T'=T'(R,C; W) such that T' < T < alpha(R, C)
T' where alpha(R,C) = min{prod r_i! r_i^{-r_i}, prod c_j! c_j^{-c_j}} N^N/N!.
In many cases, ln T' provides an asymptotically accurate estimate of ln T. The
idea of the algorithm is to express T as the expectation of the permanent of an
N x N random matrix with exponentially distributed entries and approximate the
expectation by the integral T' of an efficiently computable log-concave
function on R^{mn}. Applications to counting integer flows in graphs are also
discussed.Comment: 19 pages, bounds are sharpened, references are adde
Bounds on the permanent and some applications
We give new lower and upper bounds on the permanent of a doubly stochastic
matrix. Combined with previous work, this improves on the deterministic
approximation factor for the permanent.
We also give a combinatorial application of the lower bound, proving S.
Friedland's "Asymptotic Lower Matching Conjecture" for the monomer-dimer
problem
A permanent formula for the Jones polynomial
The permanent of a square matrix is defined in a way similar to the
determinant, but without using signs. The exact computation of the permanent is
hard, but there are Monte-Carlo algorithms that can estimate general
permanents. Given a planar diagram of a link L with crossings, we define a
7n by 7n matrix whose permanent equals to the Jones polynomial of L. This
result accompanied with recent work of Freedman, Kitaev, Larson and Wang
provides a Monte-Carlo algorithm to any decision problem belonging to the class
BQP, i.e. such that it can be computed with bounded error in polynomial time
using quantum resources.Comment: To appear in Advances in Applied Mathematic
A polynomial-time approximation algorithm for the number of k-matchings in bipartite graphs
We show that the number of -matching in a given undirected graph
is equal to the number of perfect matching of the corresponding graph
on an even number of vertices divided by a suitable factor.
If is bipartite then one can construct a bipartite .
For bipartite graphs this result implies that the number of -matching has
a polynomial-time approximation algorithm. The above results are extended to
permanents and hafnians of corresponding matrices.Comment: 6 page
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