  We propose a framework for Semi-Supervised Active Clustering framework
(SSAC), where the learner is allowed to interact with a domain expert, asking
whether two given instances belong to the same cluster or not. We study the
query and computational complexity of clustering in this framework. We consider
a setting where the expert conforms to a center-based clustering with a notion
of margin. We show that there is a trade off between computational complexity
and query complexity; We prove that for the case of $k$-means clustering (i.e.,
when the expert conforms to a solution of $k$-means), having access to
relatively few such queries allows efficient solutions to otherwise NP hard
problems.
  In particular, we provide a probabilistic polynomial-time (BPP) algorithm for
clustering in this setting that asks $O\big(k^2\log k + k\log n)$ same-cluster
queries and runs with time complexity $O\big(kn\log n)$ (where $k$ is the
number of clusters and $n$ is the number of instances). The algorithm succeeds
with high probability for data satisfying margin conditions under which,
without queries, we show that the problem is NP hard. We also prove a lower
bound on the number of queries needed to have a computationally efficient
clustering algorithm in this setting.
