  In this paper we study the problem of approximately releasing the cut
function of a graph while preserving differential privacy, and give new
algorithms (and new analyses of existing algorithms) in both the interactive
and non-interactive settings.
  Our algorithms in the interactive setting are achieved by revisiting the
problem of releasing differentially private, approximate answers to a large
number of queries on a database. We show that several algorithms for this
problem fall into the same basic framework, and are based on the existence of
objects which we call iterative database construction algorithms. We give a new
generic framework in which new (efficient) IDC algorithms give rise to new
(efficient) interactive private query release mechanisms. Our modular analysis
simplifies and tightens the analysis of previous algorithms, leading to
improved bounds. We then give a new IDC algorithm (and therefore a new private,
interactive query release mechanism) based on the Frieze/Kannan low-rank matrix
decomposition. This new release mechanism gives an improvement on prior work in
a range of parameters where the size of the database is comparable to the size
of the data universe (such as releasing all cut queries on dense graphs).
  We also give a non-interactive algorithm for efficiently releasing private
synthetic data for graph cuts with error O(|V|^{1.5}). Our algorithm is based
on randomized response and a non-private implementation of the SDP-based,
constant-factor approximation algorithm for cut-norm due to Alon and Naor.
Finally, we give a reduction based on the IDC framework showing that an
efficient, private algorithm for computing sufficiently accurate rank-1 matrix
approximations would lead to an improved efficient algorithm for releasing
private synthetic data for graph cuts. We leave finding such an algorithm as
our main open problem.
