Document Open Access Logo

On the Complexity of Community-Aware Network Sparsification

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

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2024.60.pdf
  • Filesize: 0.9 MB
  • 18 pages

Document Identifiers

Related Versions

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

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

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].

References

  1. Takanori Akiyama, Takao Nishizeki, and Nobuji Saito. NP-completeness of the Hamiltonian cycle problem for bipartite graphs. Journal of Information Processing, 3(2):73-76, 1980. Google Scholar
  2. Dana Angluin, James Aspnes, and Lev Reyzin. Network construction with subgraph connectivity constraints. Journal of Combinatorial Optimization, 29(2):418-432, 2015. Google Scholar
  3. Catriel Beeri, Ronald Fagin, David Maier, and Mihalis Yannakakis. On the desirability of acyclic database schemes. Journal of the ACM, 30(3):479-513, 1983. Google Scholar
  4. Ulrik Brandes, Sabine Cornelsen, Barbara Pampel, and Arnaud Sallaberry. Path-based supports for hypergraphs. Journal of Discrete Algorithms, 14:248-261, 2012. Google Scholar
  5. Kevin Buchin, Marc J. van Kreveld, Henk Meijer, Bettina Speckmann, and Kevin Verbeek. On planar supports for hypergraphs. Journal of Graph Algorithms and Applications, 15(4):533-549, 2011. Google Scholar
  6. Chandra Chekuri, Thapanapong Rukkanchanunt, and Chao Xu. On element-connectivity preserving graph simplification. In Proceedings of the 23rd Annual European Symposium on Algorithms (ESA '15), volume 9294 of Lecture Notes in Computer Science, pages 313-324. Springer, 2015. Google Scholar
  7. Jiehua Chen, Christian Komusiewicz, Rolf Niedermeier, Manuel Sorge, Ondrej Suchy, and Mathias Weller. Polynomial-time data reduction for the subset interconnection design problem. SIAM Journal on Discrete Mathematics, 29(1):1-25, 2015. Google Scholar
  8. Gregory V. Chockler, Roie Melamed, Yoav Tock, and Roman Vitenberg. Constructing scalable overlays for pub-sub with many topics. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Principles of Distributed Computing (PODC '07), pages 109-118. ACM, 2007. Google Scholar
  9. Nathann Cohen, Frédéric Havet, Dorian Mazauric, Ignasi Sau, and Rémi Watrigant. Complexity dichotomies for the minimum F-overlay problem. Journal of Discrete Algorithms, 52-53:133-142, 2018. Google Scholar
  10. Vincent Conitzer, Jonathan Derryberry, and Tuomas Sandholm. Combinatorial auctions with structured item graphs. In Deborah L. McGuinness and George Ferguson, editors, Proceedings of the Nineteenth National Conference on Artificial Intelligence (AAAI '04), pages 212-218. AAAI Press / The MIT Press, 2004. Google Scholar
  11. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  12. Rodney G. Downey and Michael Ralph Fellows. Fundamentals of Parameterized Complexity. Springer Science & Business Media, 2013. Google Scholar
  13. Ding-Zhu Du and Zevi Miller. Matroids and subset interconnection design. SIAM Journal on Discrete Mathematics, 1(4):416-424, 1988. Google Scholar
  14. Pierre Duchet. Propriete de Helly et problemes de representation. Colloque International CNRS, Problemes Conbinatoires et Theorie du Graphs, 260:117-118, 1978. Google Scholar
  15. Hongbing Fan, Christian Hundt, Yu-Liang Wu, and Jason Ernst. Algorithms and implementation for interconnection graph problem. In Proceedings of the Second International Conference on Combinatorial Optimization and Applications (COCOA '08), volume 5165 of Lecture Notes in Computer Science, pages 201-210. Springer, 2008. Google Scholar
  16. Claude Flament. Hypergraphes arborés. Discrete Mathematics, 21(3):223-227, 1978. Google Scholar
  17. Till Fluschnik and Leon Kellerhals. Placing green bridges optimally, with a multivariate analysis. Theory of Computing Systems, 2024. To appear. Google Scholar
  18. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  19. Aristides Gionis, Polina Rozenshtein, Nikolaj Tatti, and Evimaria Terzi. Community-aware network sparsification. In Proceedings of the 2017 SIAM International Conference on Data Mining (SDM '17), pages 426-434. SIAM, 2017. Google Scholar
  20. Nili Guttmann-Beck, Roni Rozen, and Michal Stern. Vertices removal for feasibility of clustered spanning trees. Discrete Applied Mathematics, 296:68-84, 2021. Google Scholar
  21. Nili Guttmann-Beck, Zeev Sorek, and Michal Stern. Clustered spanning tree - conditions for feasibility. Discrete Mathematics & Theoretical Computer Science, 21(1), 2019. Google Scholar
  22. Maike Herkenrath, Till Fluschnik, Francesco Grothe, and Leon Kellerhals. Placing green bridges optimally, with habitats inducing cycles. In Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, (IJCAI '22), pages 3825-3831. ijcai.org, 2022. Google Scholar
  23. Emanuel Herrendorf. On the complexity of community-aware network sparsification. Master’s thesis, Philipps-Universität Marburg, 2022. URL: https://www.fmi.uni-jena.de/fmi_femedia/24200/master-emanuel-pdf.pdf.
  24. Jun Hosoda, Juraj Hromkovic, Taisuke Izumi, Hirotaka Ono, Monika Steinová, and Koichi Wada. On the approximability and hardness of minimum topic connected overlay and its special instances. Theoretical Computer Science, 429:144-154, 2012. Google Scholar
  25. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. Google Scholar
  26. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. Google Scholar
  27. David S. Johnson and Henry O. Pollak. Hypergraph planarity and the complexity of drawing venn diagrams. Journal of Graph Theory, 11(3):309-325, 1987. Google Scholar
  28. Boris Klemz, Tamara Mchedlidze, and Martin Nöllenburg. Minimum tree supports for hypergraphs and low-concurrency Euler diagrams. In Proceedings of the 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT '14), volume 8503 of Lecture Notes in Computer Science, pages 265-276. Springer, 2014. Google Scholar
  29. Ephraim Korach and Michal Stern. The clustering matroid and the optimal clustering tree. Mathematical Programming, 98(1-3):385-414, 2003. Google Scholar
  30. Ephraim Korach and Michal Stern. The complete optimal stars-clustering-tree problem. Discrete Applied Mathematics, 156(4):444-450, 2008. Google Scholar
  31. Gerd Lindner, Christian L. Staudt, Michael Hamann, Henning Meyerhenke, and Dorothea Wagner. Structure-preserving sparsification of social networks. In Proceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM '15), pages 448-454. ACM, 2015. Google Scholar
  32. Natsu Nakajima, Morihiro Hayashida, Jesper Jansson, Osamu Maruyama, and Tatsuya Akutsu. Determining the minimum number of protein-protein interactions required to support known protein complexes. PloS one, 13(4):e0195545, 2018. Google Scholar
  33. Peter J Slater. A characterization of soft hypergraphs. Canadian Mathematical Bulletin, 21(3):335-337, 1978. Google Scholar
  34. Robert Endre Tarjan and Mihalis Yannakakis. Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on Computing, 13(3):566-579, 1984. Google Scholar
  35. Fang Zhou, Sébastien Mahler, and Hannu Toivonen. Network simplification with minimal loss of connectivity. In Proceedings of the 10th IEEE International Conference on Data Mining (ICDM '10), pages 659-668. IEEE Computer Society, 2010. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact