Document Open Access Logo

Query Complexity of Stochastic Minimum Vertex Cover

Authors Mahsa Derakhshan, Mohammad Saneian, Zhiyang Xun

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2025.41.pdf
  • Filesize: 0.67 MB
  • 12 pages

Document Identifiers

Author Details

Mahsa Derakhshan
  • Northeastern University, Boston, MA, USA
Mohammad Saneian
  • Northeastern University, Boston, MA, USA
Zhiyang Xun
  • University of Texas at Austin, TX, USA

Funding

  • Xun, Zhiyang: Supported by NSF Grant CCF-2312573 and a Simons Investigator Award (#409864, David Zuckerman).

Acknowledgements

The authors thank Sanjeev Khanna for countless insightful discussions and help.

Cite As Get BibTex

Mahsa Derakhshan, Mohammad Saneian, and Zhiyang Xun. Query Complexity of Stochastic Minimum Vertex Cover. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 41:1-41:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.41

Abstract

We study the stochastic minimum vertex cover problem for general graphs. In this problem, we are given a graph G = (V, E) and an existence probability p_e for each edge e ∈ E. Edges of G are realized (or exist) independently with these probabilities, forming the realized subgraph 𝒢. The existence of an edge in 𝒢 can only be verified using edge queries. The goal of this problem is to find a near-optimal vertex cover of 𝒢 using a small number of queries.
Previous work by Derakhshan, Durvasula, and Haghtalab [STOC 2023] established the existence of 1.5 + ε approximation algorithms for this problem with O(n/ε) queries. They also show that, under mild correlation among edge realizations, beating this approximation ratio requires querying a subgraph of size Ω(n ⋅ RS(n)). Here, RS(n) refers to Ruzsa-Szemerédi Graphs and represents the largest number of induced edge-disjoint matchings of size Θ(n) in an n-vertex graph.
In this work, we design a simple algorithm for finding a (1 + ε) approximate vertex cover by querying a subgraph of size O(n ⋅ RS(n)) for any absolute constant ε > 0. Our algorithm can tolerate up to O(n ⋅ RS(n)) correlated edges, hence effectively completing our understanding of the problem under mild correlation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Sublinear algorithms
  • Vertex cover
  • Query complexity

Metrics

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

References

  1. Sepehr Assadi. A two-pass (conditional) lower bound for semi-streaming maximum matching. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 708-742. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.32.
  2. Sepehr Assadi, Soheil Behnezhad, Sanjeev Khanna, and Huan Li. On regularity lemma and barriers in streaming and dynamic matching. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 131-144. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585110.
  3. Sepehr Assadi and Aaron Bernstein. Towards a unified theory of sparsification for matching problems. In Jeremy T. Fineman and Michael Mitzenmacher, editors, 2nd Symposium on Simplicity in Algorithms, SOSA 2019, January 8-9, 2019, San Diego, CA, USA, volume 69 of OASIcs, pages 11:1-11:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/OASICS.SOSA.2019.11.
  4. Sepehr Assadi and Sanjeev Khanna. Improved bounds for fully dynamic matching via ordered ruzsa-szemerédi graphs. CoRR, abs/2406.13573, 2024. URL: https://doi.org/10.48550/arXiv.2406.13573.
  5. Sepehr Assadi, Sanjeev Khanna, and Yang Li. The stochastic matching problem: Beating half with a non-adaptive algorithm. In Constantinos Daskalakis, Moshe Babaioff, and Hervé Moulin, editors, Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, Cambridge, MA, USA, June 26-30, 2017, pages 99-116. ACM, 2017. URL: https://doi.org/10.1145/3033274.3085146.
  6. Sepehr Assadi, Sanjeev Khanna, and Yang Li. The stochastic matching problem with (very) few queries. ACM Trans. Economics and Comput., 7(3):16:1-16:19, 2019. URL: https://doi.org/10.1145/3355903.
  7. Soheil Behnezhad, Avrim Blum, and Mahsa Derakhshan. Stochastic vertex cover with few queries. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 1808-1846. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.73.
  8. Soheil Behnezhad and Mahsa Derakhshan. Stochastic weighted matching: (stochastic weighted matching: (1-ε) approximation. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1392-1403. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00131.
  9. Soheil Behnezhad, Mahsa Derakhshan, Alireza Farhadi, MohammadTaghi Hajiaghayi, and Nima Reyhani. Stochastic matching on uniformly sparse graphs. In Dimitris Fotakis and Evangelos Markakis, editors, Algorithmic Game Theory - 12th International Symposium, SAGT 2019, Athens, Greece, September 30 - October 3, 2019, Proceedings, volume 11801 of Lecture Notes in Computer Science, pages 357-373. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-30473-7_24.
  10. Soheil Behnezhad, Mahsa Derakhshan, and MohammadTaghi Hajiaghayi. Stochastic matching with few queries: (1-ε) approximation. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 1111-1124. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384340.
  11. Soheil Behnezhad and Alma Ghafari. Fully dynamic matching and ordered ruzsa-szemerédi graphs. In 65st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024, 2024. Google Scholar
  12. Soheil Behnezhad and Nima Reyhani. Almost optimal stochastic weighted matching with few queries. In Éva Tardos, Edith Elkind, and Rakesh Vohra, editors, Proceedings of the 2018 ACM Conference on Economics and Computation, Ithaca, NY, USA, June 18-22, 2018, pages 235-249. ACM, 2018. URL: https://doi.org/10.1145/3219166.3219226.
  13. Avrim Blum, John P. Dickerson, Nika Haghtalab, Ariel D. Procaccia, Tuomas Sandholm, and Ankit Sharma. Ignorance is almost bliss: Near-optimal stochastic matching with few queries. In Tim Roughgarden, Michal Feldman, and Michael Schwarz, editors, Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC '15, Portland, OR, USA, June 15-19, 2015, pages 325-342. ACM, 2015. URL: https://doi.org/10.1145/2764468.2764479.
  14. Mahsa Derakhshan, Naveen Durvasula, and Nika Haghtalab. Stochastic minimum vertex cover in general graphs: A 3/2-approximation. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 242-253. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585230.
  15. Mahsa Derakhshan and Mohammad Saneian. Query efficient weighted stochastic matching. CoRR, abs/2311.08513, 2023. URL: https://doi.org/10.48550/arXiv.2311.08513.
  16. Shaddin Dughmi, Yusuf Hakan Kalayci, and Neel Patel. On sparsification of stochastic packing problems. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 51:1-51:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.51.
  17. Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 474-483, 2002. URL: https://doi.org/10.1145/509907.509977.
  18. Jacob Fox. A new proof of the graph removal lemma. Annals of Mathematics, pages 561-579, 2011. Google Scholar
  19. Jacob Fox, Hao Huang, and Benny Sudakov. On graphs decomposable into induced matchings of linear sizes. Bulletin of the London Mathematical Society, 49(1):45-57, 2017. Google Scholar
  20. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 468-485. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.41.
  21. Michel X. Goemans and Jan Vondrák. Covering minimum spanning trees of random subgraphs. In J. Ian Munro, editor, Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, New Orleans, Louisiana, USA, January 11-14, 2004, pages 934-941. SIAM, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982933.
  22. Michel X. Goemans and Jan Vondrák. Covering minimum spanning trees of random subgraphs. Random Struct. Algorithms, 29(3):257-276, 2006. URL: https://doi.org/10.1002/RSA.20115.
  23. Imre Z Ruzsa and Endre Szemerédi. Triple systems with no six points carrying three triangles. Combinatorics (Keszthely, 1976), Coll. Math. Soc. J. Bolyai, 18(939-945):2, 1978. Google Scholar
  24. Jan Vondrák. Shortest-path metric approximation for random subgraphs. Random Struct. Algorithms, 30(1-2):95-104, 2007. URL: https://doi.org/10.1002/RSA.20150.
  25. Yutaro Yamaguchi and Takanori Maehara. Stochastic packing integer programs with few queries. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 293-310. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.21.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact