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Description:
Given an undirected graph G, the Densest k-subgraph problem (DkS) asks to compute a set S ⊂ V of cardinality |S| ≤ 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 𝒪(n^{1/4 + ε}) approximation in time n^{𝒪(1/ε)}, for any ε > 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.