  Given an undirected graph $G$, the Densest $k$-subgraph problem (DkS) asks to
compute a set $S \subset V$ of cardinality $\left\lvert S\right\rvert \leq k$
such that the weight of edges inside $S$ is maximized. This is a fundamental
NP-hard problem whose approximability, inspite of many decades of research, is
yet to be settled. The current best known approximation algorithm due to
Bhaskara et al. (2010) computes a $\mathcal{O}\left({n^{1/4 +
\epsilon}}\right)$ approximation in time
$n^{\mathcal{O}\left(1/\epsilon\right)}$, for any $\epsilon > 0$.
  We ask what are some "easier" instances of this problem? We propose some
natural semi-random models of instances with a planted dense subgraph, and
study approximation algorithms for computing the densest subgraph in them.
These models are inspired by the semi-random models of instances studied for
various other graph problems such as the independent set problem, graph
partitioning problems etc. For a large range of parameters of these models, we
get significantly better approximation factors for the Densest $k$-subgraph
problem. Moreover, our algorithm recovers a large part of the planted solution.
