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On the Generalized Mean Densest Subgraph Problem: Complexity and Algorithms

Authors Karthekeyan Chandrasekaran , Chandra Chekuri , Manuel R. Torres , Weihao Zhu

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Densest subgraph problem
  • Hardness of approximation
  • Approximation algorithms

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Abstract

Dense subgraph discovery is an important problem in graph mining and network analysis with several applications. Two canonical polynomial-time solvable problems here are to find a maxcore (subgraph of maximum min degree) and to find a densest subgraph (subgraph of maximum average degree). Both of these problems can be solved in polynomial time. Veldt, Benson, and Kleinberg [Veldt et al., 2021] introduced the generalized p-mean densest subgraph problem which captures the maxcore problem when p = -∞ and the densest subgraph problem when p = 1. They observed that for p ≥ 1, the objective function is supermodular and hence the problem can be solved in polynomial time. In this work, we focus on the p-mean densest subgraph problem for p ∈ (-∞, 1). We prove that for every p ∈ (-∞,1), the problem is NP-hard, thus resolving an open question from [Veldt et al., 2021]. We also show that for every p ∈ (0,1), the weighted version of the problem is APX-hard. On the algorithmic front, we describe two simple 1/2-approximation algorithms for every p ∈ (-∞, 1). We complement the approximation algorithms by exhibiting non-trivial instances on which the algorithms simultaneously achieve an approximation factor of at most 1/2.

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Karthekeyan Chandrasekaran, Chandra Chekuri, Manuel R. Torres, and Weihao Zhu. On the Generalized Mean Densest Subgraph Problem: Complexity and Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.9

Author Details

Karthekeyan Chandrasekaran
  • University of Illinois, Urbana-Champaign, USA
Chandra Chekuri
  • University of Illinois, Urbana-Champaign, USA
Manuel R. Torres
  • University of Illinois, Urbana-Champaign, USA
Weihao Zhu
  • University of Illinois, Urbana-Champaign, USA

Funding

  • Chandrasekaran, Karthekeyan: partially supported by NSF grant CCF-1907937 AND CCF-2402667.
  • Chekuri, Chandra: partially supported by NSF grants CCF-1907937 and CCF-2402667.
  • Torres, Manuel R.: supported in part by fellowships from NSF and the Sloan Foundation, and NSF grant CCF-1910149.
  • Zhu, Weihao: supported by a graduate fellowship from the CS department.

Acknowledgements

We thank Sanjeev Khanna and Euiwoong Lee for pointers to [Garg et al., 2021] and [Gupta et al., 2019] on the hardness of Exact 𝓁-Cover. We thank Farouk Harb for helpful discussions. This work was done when Manuel R. Torres was a student at University of Illinois, Urbana-Champaign.

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