12,355 research outputs found
Approximating the Permanent with Fractional Belief Propagation
We discuss schemes for exact and approximate computations of permanents, and
compare them with each other. Specifically, we analyze the Belief Propagation
(BP) approach and its Fractional Belief Propagation (FBP) generalization for
computing the permanent of a non-negative matrix. Known bounds and conjectures
are verified in experiments, and some new theoretical relations, bounds and
conjectures are proposed. The Fractional Free Energy (FFE) functional is
parameterized by a scalar parameter , where
corresponds to the BP limit and corresponds to the exclusion
principle (but ignoring perfect matching constraints) Mean-Field (MF) limit.
FFE shows monotonicity and continuity with respect to . For every
non-negative matrix, we define its special value to be the
for which the minimum of the -parameterized FFE functional is
equal to the permanent of the matrix, where the lower and upper bounds of the
-interval corresponds to respective bounds for the permanent. Our
experimental analysis suggests that the distribution of varies for
different ensembles but always lies within the interval.
Moreover, for all ensembles considered the behavior of is highly
distinctive, offering an emprirical practical guidance for estimating
permanents of non-negative matrices via the FFE approach.Comment: 42 pages, 14 figure
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
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
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