3,103 research outputs found
Modified Bethe Permanent of a Nonnegative Matrix
Currently the best deterministic polynomial-time algorithm for approximating the permanent of a non-negative matrix is based on minimizing the Bethe free energy function of a certain normal factor graph (NFG). In order to improve the approximation guarantee, we propose a modified NFG with fewer cycles, but still manageable function-node complexity; we call the approximation obtained by minimizing the function of the modified normal factor graph the modified Bethe permanent. For nonnegative matrices of size 3× 3, we give a tight characterization of the modified Bethe permanent. For non-negative matrices of size n× n with n≥ 3, we present a partial characterization, along with promising numerical results. The analysis of the modified NFG is also interesting because of its tight connection to an NFG that is used for approximating a permanent-like quantity in quantum information processing. © 2020 IEEE
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
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
Degree- Bethe and Sinkhorn Permanent Based Bounds on the Permanent of a Non-negative Matrix
The permanent of a non-negative square matrix can be well approximated by
finding the minimum of the Bethe free energy functions associated with some
suitably defined factor graph; the resulting approximation to the permanent is
called the Bethe permanent. Vontobel gave a combinatorial characterization of
the Bethe permanent via degree- Bethe permanents, which is based on
degree- covers of the underlying factor graph. In this paper, we prove a
degree--Bethe-permanent-based lower bound on the permanent of a non-negative
matrix, which solves a conjecture proposed by Vontobel in [IEEE Trans. Inf.
Theory, Mar. 2013]. We also prove a degree--Bethe-permanent-based upper
bound on the permanent of a non-negative matrix. In the limit ,
these lower and upper bounds yield known Bethe-permanent-based lower and upper
bounds on the permanent of a non-negative matrix. Moreover, we prove similar
results for an approximation to the permanent known as the (scaled) Sinkhorn
permanent.Comment: submitte
The number of matchings in random graphs
We study matchings on sparse random graphs by means of the cavity method. We
first show how the method reproduces several known results about maximum and
perfect matchings in regular and Erdos-Renyi random graphs. Our main new result
is the computation of the entropy, i.e. the leading order of the logarithm of
the number of solutions, of matchings with a given size. We derive both an
algorithm to compute this entropy for an arbitrary graph with a girth that
diverges in the large size limit, and an analytic result for the entropy in
regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic
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