5 research outputs found
A deterministic approximation algorithm for computing the permanent of a 0, 1 matrix
We consider the problem of computing the permanent of a n by n matrix. For a class of matrices corresponding to constant degree expanders we construct a deterministic polynomial time approximation algorithm to within a multiplicative factor ( 1 + â)[superscript η] for arbitraryâ > 0. This is an improvement over the best known approximation factor e[superscript η] obtained in Linial, Samorodnitsky and Wigderson (2000), though the latter result was established for arbitrary non-negative matrices. Our results use a recently developed deterministic approximation algorithm for counting partial matchings of a graph (Bayati, Gamarnik, Katz, Nair and Tetali (2007)) and JerrumâVazirani method (Jerrum and Vazirani (1996)) of approximating permanent by near perfect matchings
Computing the permanent of (some) complex matrices
We present a deterministic algorithm, which, for any given 0< epsilon < 1 and
an nxn real or complex matrix A=(a_{ij}) such that | a_{ij}-1| < 0.19 for all
i, j computes the permanent of A within relative error epsilon in n^{O(ln n -ln
epsilon)} time. The method can be extended to computing hafnians and
multidimensional permanents.Comment: 12 pages, results extended to hafnians and multidimensional
permanents, minor improvement
Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs
For a graph , let be the partition function of the
monomer-dimer system defined by , where is the
number of matchings of size in . We consider graphs of bounded degree
and develop a sublinear-time algorithm for estimating at an
arbitrary value within additive error with high
probability. The query complexity of our algorithm does not depend on the size
of and is polynomial in , and we also provide a lower bound
quadratic in for this problem. This is the first analysis of a
sublinear-time approximation algorithm for a # P-complete problem. Our
approach is based on the correlation decay of the Gibbs distribution associated
with . We show that our algorithm approximates the probability
for a vertex to be covered by a matching, sampled according to this Gibbs
distribution, in a near-optimal sublinear time. We extend our results to
approximate the average size and the entropy of such a matching within an
additive error with high probability, where again the query complexity is
polynomial in and the lower bound is quadratic in .
Our algorithms are simple to implement and of practical use when dealing with
massive datasets. Our results extend to other systems where the correlation
decay is known to hold as for the independent set problem up to the critical
activity