  We construct a deterministic approximation algorithm for computing a
permanent of a $0,1$ $n$ by $n$ matrix to within a multiplicative factor
$(1+\epsilon)^n$, for arbitrary $\epsilon>0$. When the graph underlying the
matrix is a constant degree expander our algorithm runs in polynomial time
(PTAS). In the general case the running time of the algorithm is
$\exp(O(n^{2\over 3}\log^3n))$. For the class of graphs which are constant
degree expanders the first result is an improvement over the best known
approximation factor $e^n$ obtained in \cite{LinialSamorodnitskyWigderson}.
  Our results use a recently developed deterministic approximation algorithm
for counting partial matchings of a graph Bayati et al., and Jerrum-Vazirani
decomposition method.
