On the Cut-Query Complexity of Approximating Max-Cut

Authors Orestis Plevrakis, Seyoon Ragavan , S. Matthew Weinberg



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.115.pdf
  • Filesize: 0.81 MB
  • 20 pages

Document Identifiers

Author Details

Orestis Plevrakis
  • Department of Computer Science, Princeton University, NJ, USA
Seyoon Ragavan
  • Computer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology, Cambridge, MA, USA
S. Matthew Weinberg
  • Department of Computer Science, Princeton University, NJ, USA

Acknowledgements

SR would like to thank Georgy Noarov and Dmitry Paramonov for collaborating on a course project that led to this work. We would also like to thank Sepehr Assadi for helpful discussions and anonymous reviewers for feedback and suggested changes.

Cite AsGet BibTex

Orestis Plevrakis, Seyoon Ragavan, and S. Matthew Weinberg. On the Cut-Query Complexity of Approximating Max-Cut. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.115

Abstract

We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [Rubinstein et al., 2018]. Graph algorithms in this cut query model and other query models have recently been studied for various other problems such as min-cut, connectivity, bipartiteness, and triangle detection. Max-cut in the cut query model can also be viewed as a natural special case of submodular function maximization: on query S ⊆ V, the oracle returns the total weight of the cut between S and V\S. Our first main technical result is a lower bound stating that a deterministic algorithm achieving a c-approximation for any c > 1/2 requires Ω(n) queries. This uses an extension of the cut dimension to rule out approximation (prior work of [Graur et al., 2020] introducing the cut dimension only rules out exact solutions). Secondly, we provide a randomized algorithm with Õ(n) queries that finds a c-approximation for any c < 1. We achieve this using a query-efficient sparsifier for undirected weighted graphs (prior work of [Rubinstein et al., 2018] holds only for unweighted graphs). To complement these results, for most constants c ∈ (0,1], we nail down the query complexity of achieving a c-approximation, for both deterministic and randomized algorithms (up to logarithmic factors). Analogously to general submodular function maximization in the same model, we observe a phase transition at c = 1/2: we design a deterministic algorithm for global c-approximate max-cut in O(log n) queries for any c < 1/2, and show that any randomized algorithm requires Ω(n/log n) queries to find a c-approximate max-cut for any c > 1/2. Additionally, we show that any deterministic algorithm requires Ω(n²) queries to find an exact max-cut (enough to learn the entire graph).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
Keywords
  • query complexity
  • maximum cut
  • approximation algorithms
  • graph sparsification

Metrics

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

References

  1. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. 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 459-467. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.40.
  2. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Michael Benedikt, Markus Krötzsch, and Maurizio Lenzerini, editors, Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2012, Scottsdale, AZ, USA, May 20-24, 2012, pages 5-14. ACM, 2012. URL: https://doi.org/10.1145/2213556.2213560.
  3. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Spectral sparsification in dynamic graph streams. In Prasad Raghavendra, Sofya Raskhodnikova, Klaus Jansen, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings, volume 8096 of Lecture Notes in Computer Science, pages 1-10. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_1.
  4. Simon Apers, Yuval Efron, Pawel Gawrychowski, Troy Lee, Sagnik Mukhopadhyay, and Danupon Nanongkai. Cut query algorithms with star contraction. CoRR, abs/2201.05674, 2022. URL: https://arxiv.org/abs/2201.05674.
  5. Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna. Graph connectivity and single element recovery via linear and OR queries. In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms, ESA 2021, September 6-8, 2021, Lisbon, Portugal (Virtual Conference), volume 204 of LIPIcs, pages 7:1-7:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ESA.2021.7.
  6. Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM Rev., 56(2):315-334, 2014. URL: https://doi.org/10.1137/130949117.
  7. András A. Benczúr and David R. Karger. Randomized approximation schemes for cuts and flows in capacitated graphs. SIAM J. Comput., 44(2):290-319, 2015. URL: https://doi.org/10.1137/070705970.
  8. Joakim Blikstad, Jan van den Brand, Yuval Efron, Sagnik Mukhopadhyay, and Danupon Nanongkai. Nearly optimal communication and query complexity of bipartite matching. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 1174-1185. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00113.
  9. Nader H. Bshouty and Hanna Mazzawi. Toward a deterministic polynomial time algorithm with optimal additive query complexity. Theor. Comput. Sci., 417:23-35, 2012. URL: https://doi.org/10.1016/J.TCS.2011.09.005.
  10. Nader H. Bshouty and Hanna Mazzawi. On parity check (0, 1)-matrix over 𝕫_p. SIAM J. Discret. Math., 29(1):631-657, 2015. URL: https://doi.org/10.1137/120881129.
  11. Niv Buchbinder and Moran Feldman. Deterministic algorithms for submodular maximization problems. ACM Trans. Algorithms, 14(3):32:1-32:20, 2018. URL: https://doi.org/10.1145/3184990.
  12. Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. A tight linear time (1/2)-approximation for unconstrained submodular maximization. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 649-658. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/FOCS.2012.73.
  13. Deeparnab Chakrabarty, Andrei Graur, Haotian Jiang, and Aaron Sidford. Improved lower bounds for submodular function minimization. CoRR, abs/2207.04342, 2022. URL: https://doi.org/10.48550/arXiv.2207.04342.
  14. Deeparnab Chakrabarty and Hang Liao. A query algorithm for learning a spanning forest in weighted undirected graphs. In Shipra Agrawal and Francesco Orabona, editors, International Conference on Algorithmic Learning Theory, February 20-23, 2023, Singapore, volume 201 of Proceedings of Machine Learning Research, pages 259-274. PMLR, 2023. URL: https://proceedings.mlr.press/v201/chakrabarty23a.html.
  15. Sung-Soon Choi. Polynomial time optimal query algorithms for finding graphs with arbitrary real weights. In Shai Shalev-Shwartz and Ingo Steinwart, editors, Proceedings of the 26th Annual Conference on Learning Theory, volume 30 of Proceedings of Machine Learning Research, pages 797-818, Princeton, NJ, USA, 12-14 June 2013. PMLR. URL: https://proceedings.mlr.press/v30/Choi13.html.
  16. Sung-Soon Choi and Jeong Han Kim. Optimal query complexity bounds for finding graphs. In Cynthia Dwork, editor, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 749-758. ACM, 2008. URL: https://doi.org/10.1145/1374376.1374484.
  17. Marcel Kenji de Carli Silva, Nicholas J. A. Harvey, and Cristiane M. Sato. Sparse sums of positive semidefinite matrices. ACM Trans. Algorithms, 12(1):9:1-9:17, 2016. URL: https://doi.org/10.1145/2746241.
  18. Uriel Feige, Vahab S. Mirrokni, and Jan Vondrák. Maximizing non-monotone submodular functions. SIAM J. Comput., 40(4):1133-1153, 2011. URL: https://doi.org/10.1137/090779346.
  19. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115-1145, 1995. URL: https://doi.org/10.1145/227683.227684.
  20. Andrei Graur, Tristan Pollner, Vikram Ramaswamy, and S. Matthew Weinberg. New query lower bounds for submodular function minimization. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 64:1-64:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.64.
  21. Vladimir Grebinski and Gregory Kucherov. Optimal reconstruction of graphs under the additive model. Algorithmica, 28(1):104-124, 2000. URL: https://doi.org/10.1007/s004530010033.
  22. Nicholas J. A. Harvey. Matchings, matroids and submodular functions. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2008. URL: http://hdl.handle.net/1721.1/44416.
  23. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
  24. Haotian Jiang. Minimizing convex functions with integral minimizers. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 976-985. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.61.
  25. Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 561-570. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/FOCS.2014.66.
  26. Subhash Khot. On the power of unique 2-prover 1-round games. In John H. Reif, editor, Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 767-775. ACM, 2002. URL: https://doi.org/10.1145/509907.510017.
  27. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput., 37(1):319-357, 2007. URL: https://doi.org/10.1137/S0097539705447372.
  28. Troy Lee, Tongyang Li, Miklos Santha, and Shengyu Zhang. On the cut dimension of a graph. In Valentine Kabanets, editor, 36th Computational Complexity Conference, CCC 2021, July 20-23, 2021, Toronto, Ontario, Canada (Virtual Conference), volume 200 of LIPIcs, pages 15:1-15:35. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.15.
  29. Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1049-1065. IEEE Computer Society, 2015. URL: https://doi.org/10.1109/FOCS.2015.68.
  30. Sagnik Mukhopadhyay and Danupon Nanongkai. Weighted min-cut: sequential, cut-query, and streaming algorithms. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 496-509. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384334.
  31. Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing exact minimum cuts without knowing the graph. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 39:1-39:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.39.
  32. Xiaoming Sun, David P. Woodruff, Guang Yang, and Jialin Zhang. Querying a matrix through matrix-vector products. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 94:1-94:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.94.