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Planted Models for the Densest k-Subgraph Problem

Authors Yash Khanna, Anand Louis

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Author Details

Yash Khanna
  • Indian Institute of Science, Bangalore, India
Anand Louis
  • Indian Institute of Science, Bangalore, India

Funding

  • Louis, Anand: AL was supported in part by SERB Award ECR/2017/003296 and a Pratiksha Trust Young Investigator Award.

Acknowledgements

We thank Rakesh Venkat for helpful discussions. We also thank the anonymous reviewers for their suggestions and comments on earlier versions of this paper.

Cite As Get BibTex

Yash Khanna and Anand Louis. Planted Models for the Densest k-Subgraph Problem. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.FSTTCS.2020.27

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Semidefinite programming
  • Theory of computation → Discrete optimization
  • Theory of computation → Graph algorithms analysis
Keywords
  • Densest k-Subgraph
  • Semi-Random models
  • Planted Models
  • Semidefinite Programming
  • Approximation Algorithms
  • Beyond Worst Case Analysis

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