Planted Models for the Densest $k$-Subgraph Problem

Abstract

Given an undirected graph $G$, the Densest $k$-subgraph problem (DkS) asks to compute a set $S\subset V$ of cardinality $\left|S\right|\le k$ 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.

31 pages. To appear in FSTTCS 2020