Document Open Access Logo

On Deleting Vertices to Reduce Density in Graphs and Supermodular Functions

Authors Karthekeyan Chandrasekaran , Chandra Chekuri, Shubhang Kulkarni

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.43.pdf
  • Filesize: 0.94 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Combinatorial Optimization
  • Approximation Algorithms
  • Randomized Algorithms
  • Hardness of Approximation
  • Densest Subgraph
  • Supermodular Functions
  • Submodular Set Cover

Metrics

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

Abstract

We consider deletion problems in graphs and supermodular functions where the goal is to reduce density. In Graph Density Deletion (GraphDD), we are given a graph G = (V,E) with non-negative vertex costs and a non-negative parameter ρ ≥ 0 and the goal is to remove a minimum cost subset S of vertices such that the densest subgraph in G-S has density at most ρ. This problem has an underlying matroidal structure and generalizes several classical problems such as vertex cover, feedback vertex set, and pseudoforest deletion set for appropriately chosen ρ ≤ 1 and all of these classical problems admit a 2-approximation. In sharp contrast, we prove that for every fixed integer ρ > 1, GraphDD is hard to approximate to within a logarithmic factor via a reduction from SetCover, thus showing a phase transition phenomenon. Next, we investigate a generalization of GraphDD to monotone supermodular functions, termed Supermodular Density Deletion (SupmodDD). In SupmodDD, we are given a monotone supermodular function f:2^V → ℤ_{≥0} via an evaluation oracle with element costs and a non-negative integer ρ ≥ 0 and the goal is remove a minimum cost subset S ⊆ V such that the densest subset according to f in V-S has density at most ρ. We show that SupmodDD is approximation equivalent to the well-known Submodular Cover problem; this implies a tight logarithmic approximation and hardness for SupmodDD; it also implies a logarithmic approximation for GraphDD, thus matching our inapproximability bound. Motivated by these hardness results, we design bicriteria approximation algorithms for both GraphDD and SupmodDD.

Cite As Get BibTex

Karthekeyan Chandrasekaran, Chandra Chekuri, and Shubhang Kulkarni. On Deleting Vertices to Reduce Density in Graphs and Supermodular Functions. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 43:1-43:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.43

Author Details

Karthekeyan Chandrasekaran
  • Grainger College of Engineering, University of Illinois, Urbana-Champaign, IL, USA
Chandra Chekuri
  • Grainger College of Engineering, University of Illinois, Urbana-Champaign, IL, USA
Shubhang Kulkarni
  • Grainger College of Engineering, University of Illinois, Urbana-Champaign, IL, USA

Funding

This work was supported in part by NSF grant CCF-2402667.

Acknowledgements

Shubhang Kulkarni thanks Kishen Gowda for engaging in preliminary discussions about approximations for feedback vertex set and pseudoforest deletion set. We thank anonymous reviewers for pointing out an error, and for suggesting improvements to a previous version of this work.

References

  1. Vineet Bafna, Piotr Berman, and Toshihiro Fujito. Constant ratio approximations of the weighted feedback vertex set problem for undirected graphs. In Algorithms and Computations, pages 142-151, 1995. URL: https://doi.org/10.1007/BFB0015417.
  2. Cristina Bazgan, André Nichterlein, and Sofia Vazquez Alferez. Destroying Densest Subgraphs Is Hard. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024), pages 6:1-6:17, 2024. URL: https://doi.org/10.4230/LIPICS.SWAT.2024.6.
  3. Ann Becker and Dan Geiger. Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence, 83:167-188, 1996. URL: https://doi.org/10.1016/0004-3702(95)00004-6.
  4. Digvijay Boob, Yu Gao, Richard Peng, Saurabh Sawlani, Charalampos Tsourakakis, Di Wang, and Junxing Wang. Flowless: Extracting densest subgraphs without flow computations. In Proceedings of The Web Conference 2020, pages 573-583, 2020. URL: https://doi.org/10.1145/3366423.3380140.
  5. Karthekeyan Chandrasekaran, Chandra Chekuri, Samuel Fiorini, Shubhang Kulkarni, and Stefan Weltge. Polyhedral aspects of feedback vertex set and pseudoforest deletion set. Mathematical Programming, pages 1-48, 2025. Google Scholar
  6. Karthekeyan Chandrasekaran, Chandra Chekuri, and Shubhang Kulkarni. On deleting vertices to reduce density in graphs and supermodular functions, 2025. URL: https://doi.org/10.48550/arXiv.2503.08828.
  7. Moses Charikar. Greedy Approximation Algorithms for Finding Dense Components in a Graph. In Approximation Algorithms for Combinatorial Optimization, pages 84-95, 2000. URL: https://doi.org/10.1007/3-540-44436-X_10.
  8. Chandra Chekuri, Aleksander Bjørn Christiansen, Jacob Holm, Ivor van der Hoog, Kent Quanrud, Eva Rotenberg, and Chris Schwiegelshohn. Adaptive out-orientations with applications. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3062-3088, 2024. Google Scholar
  9. Chandra Chekuri, Kent Quanrud, and Manuel R Torres. Densest subgraph: Supermodularity, iterative peeling, and flow. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1531-1555, 2022. URL: https://doi.org/10.1137/1.9781611977073.64.
  10. Fabián A. Chudak, Michel X. Goemans, Dorit S. Hochbaum, and David P. Williamson. A primal–dual interpretation of two 2-approximation algorithms for the feedback vertex set problem in undirected graphs. Operations Research Letters, 22(4):111-118, 1998. URL: https://doi.org/10.1016/S0167-6377(98)00021-2.
  11. William H Cunningham. Optimal attack and reinforcement of a network. Journal of the ACM (JACM), 32(3):549-561, 1985. URL: https://doi.org/10.1145/3828.3829.
  12. Laxman Dhulipala, Quanquan C Liu, Sofya Raskhodnikova, Jessica Shi, Julian Shun, and Shangdi Yu. Differential privacy from locally adjustable graph algorithms: k-core decomposition, low out-degree ordering, and densest subgraphs. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 754-765, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00077.
  13. Michael Dinitz, Satyen Kale, Silvio Lattanzi, and Sergei Vassilvitskii. Almost tight bounds for differentially private densest subgraph. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2908-2950, 2025. URL: https://doi.org/10.1137/1.9781611978322.94.
  14. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC, pages 624-633, 2014. URL: https://doi.org/10.1145/2591796.2591884.
  15. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  16. A. Frank. Connections in combinatorial optimization. Oxford University Press, Oxford, 2011. Google Scholar
  17. Satoru Fujishige. Lexicographically optimal base of a polymatroid with respect to a weight vector. Mathematics of Operations Research, 5(2):186-196, 1980. URL: https://doi.org/10.1287/MOOR.5.2.186.
  18. Toshihiro Fujito. Approximating node-deletion problems for matroidal properties. Journal of Algorithms, 31(1):211-227, 1999. URL: https://doi.org/10.1006/JAGM.1998.0995.
  19. A. V. Goldberg. Finding a maximum density subgraph. Technical report, University of California at Berkeley, USA, 1984. Google Scholar
  20. Elfarouk Harb, Kent Quanrud, and Chandra Chekuri. Faster and scalable algorithms for densest subgraph and decomposition. In Advances in Neural Information Processing Systems, 2022. Google Scholar
  21. Elfarouk Harb, Kent Quanrud, and Chandra Chekuri. Convergence to lexicographically optimal base in a (contra)polymatroid and applications to densest subgraph and tree packing. In 31st Annual European Symposium on Algorithms (ESA), pages 56:1-56:17, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.56.
  22. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2- ε. Journal of Computer and System Sciences, 74(3):335-349, 2008. Google Scholar
  23. Tommaso Lanciano, Atsushi Miyauchi, Adriano Fazzone, and Francesco Bonchi. A survey on the densest subgraph problem and its variants. ACM Comput. Surv., 56(8), April 2024. URL: https://doi.org/10.1145/3653298.
  24. Audrey Lee and Ileana Streinu. Pebble game algorithms and sparse graphs. Discrete Mathematics, 308(8):1425-1437, April 2008. URL: https://doi.org/10.1016/J.DISC.2007.07.104.
  25. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is np-complete. Journal of Computer and System Sciences, 20(2):219-230, April 1980. URL: https://doi.org/10.1016/0022-0000(80)90060-4.
  26. Ta Duy Nguyen and Alina Ene. Multiplicative weights update, area convexity and random coordinate descent for densest subgraph problems. In Proceedings of the 41st International Conference on Machine Learning (ICML), 2024. Google Scholar
  27. Jean-Claude Picard and Maurice Queyranne. A network flow solution to some nonlinear 0-1 programming problems, with applications to graph theory. Networks, 12(2):141-159, 1982. URL: https://doi.org/10.1002/NET.3230120206.
  28. Saurabh Sawlani and Junxing Wang. Near-optimal fully dynamic densest subgraph. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 181-193, 2020. URL: https://doi.org/10.1145/3357713.3384327.
  29. Shuichi Ueno, Yoji Kajitani, and Shin’ya Gotoh. On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three. Discrete Mathematics, 72(1):355-360, December 1988. URL: https://doi.org/10.1016/0012-365X(88)90226-9.
  30. Nate Veldt, Austin R Benson, and Jon Kleinberg. The generalized mean densest subgraph problem. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pages 1604-1614, 2021. URL: https://doi.org/10.1145/3447548.3467398.
  31. Laurence A Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2(4):385-393, 1982. URL: https://doi.org/10.1007/BF02579435.
  32. Michał Włodarczyk. Losing treewidth in the presence of weights. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3743-3761, 2025. 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