  Motivated by applications in crowdsourced entity resolution in database,
signed edge prediction in social networks and correlation clustering, Mazumdar
and Saha [NIPS 2017] proposed an elegant theoretical model for studying
clustering with a faulty oracle. In this model, given a set of $n$ items which
belong to $k$ unknown groups (or clusters), our goal is to recover the clusters
by asking pairwise queries to an oracle. This oracle can answer the query that
``do items $u$ and $v$ belong to the same cluster?''. However, the answer to
each pairwise query errs with probability $\varepsilon$, for some
$\varepsilon\in(0,\frac12)$. Mazumdar and Saha provided two algorithms under
this model: one algorithm is query-optimal while time-inefficient (i.e.,
running in quasi-polynomial time), the other is time efficient (i.e., in
polynomial time) while query-suboptimal. Larsen, Mitzenmacher and Tsourakakis
[WWW 2020] then gave a new time-efficient algorithm for the special case of $2$
clusters, which is query-optimal if the bias $\delta:=1-2\varepsilon$ of the
model is large. It was left as an open question whether one can obtain a
query-optimal, time-efficient algorithm for the general case of $k$ clusters
and other regimes of $\delta$.
  In this paper, we make progress on the above question and provide a
time-efficient algorithm with nearly-optimal query complexity (up to a factor
of $O(\log^2 n)$) for all constant $k$ and any $\delta$ in the regime when
information-theoretic recovery is possible. Our algorithm is built on a
connection to the stochastic block model.
