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Planted Models for the Densest k-Subgraph Problem

Authors Yash Khanna, Anand Louis

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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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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.

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

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.

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