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On the Complexity of Community-Aware Network Sparsification

Authors Emanuel Herrendorf, Christian Komusiewicz , Nils Morawietz , Frank Sommer

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Author Details

Emanuel Herrendorf
  • Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany
Christian Komusiewicz
  • Friedrich Schiller University Jena, Institute of Computer Science, Germany
Nils Morawietz
  • Friedrich Schiller University Jena, Institute of Computer Science, Germany
Frank Sommer
  • Friedrich Schiller University Jena, Institute of Computer Science, Germany

Funding

  • Morawietz, Nils: Partially supported by the DFG, project OPERAH (KO 3669/5-1).
  • Sommer, Frank: Partially supported by the DFG, project EAGR (KO 3669/6-1)

Acknowledgements

Some of the results of this work are also contained in the first author’s Masters thesis [Herrendorf, 2022].

Cite As Get BibTex

Emanuel Herrendorf, Christian Komusiewicz, Nils Morawietz, and Frank Sommer. On the Complexity of Community-Aware Network Sparsification. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 60:1-60:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.60

Abstract

In the NP-hard Π-Network Sparsification problem, we are given an edge-weighted graph G, a collection 𝒞 of c subsets of V(G), called communities, and two numbers 𝓁 and b, and the question is whether there exists a spanning subgraph G' of G with at most 𝓁 edges of total weight at most b such that G'[C] fulfills Π for each community C ∈ 𝒞. We study the fine-grained and parameterized complexity of two special cases of this problem: Connectivity NWS where Π is the connectivity property and Stars NWS, where Π is the property of having a spanning star. 
First, we provide a tight 2^Ω(n²+c)-time running time lower bound based on the ETH for both problems, where n is the number of vertices in G even if all communities have size at most 4, G is a clique, and every edge has unit weight. For the connectivity property, the unit weight case with G being a clique is the well-studied problem of computing a hypergraph support with a minimum number of edges. We then study the complexity of both problems parameterized by the feedback edge number t of the solution graph G'. For Stars NWS, we present an XP-algorithm for t answering an open question by Korach and Stern [Discret. Appl. Math. '08] who asked for the existence of polynomial-time algorithms for t = 0. In contrast, we show for Connectivity NWS that known polynomial-time algorithms for t = 0 [Korach and Stern, Math. Program. '03; Klemz et al., SWAT '14] cannot be extended to larger values of t by showing NP-hardness for t = 1.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Hypergraphs
Keywords
  • parameterized complexity
  • hypergraph support
  • above guarantee parameterization
  • exponential-time-hypothesis

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