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Learning-Augmented Maximum Independent Set

Authors Vladimir Braverman, Prathamesh Dharangutte, Vihan Shah, Chen Wang

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  • Theory of computation → Graph algorithms analysis
  • Computing methodologies → Machine learning
Keywords
  • Learning-augmented algorithms
  • maximum independent set
  • graph algorithms

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Abstract

We study the Maximum Independent Set (MIS) problem on general graphs within the framework of learning-augmented algorithms. The MIS problem is known to be NP-hard and is also NP-hard to approximate to within a factor of n^(1-δ) for any δ > 0. We show that we can break this barrier in the presence of an oracle obtained through predictions from a machine learning model that answers vertex membership queries for a fixed MIS with probability 1/2+ε. In the first setting we consider, the oracle can be queried once per vertex to know if a vertex belongs to a fixed MIS, and the oracle returns the correct answer with probability 1/2 + ε. Under this setting, we show an algorithm that obtains an Õ((√Δ)/ε)-approximation in O(m) time where Δ is the maximum degree of the graph. In the second setting, we allow multiple queries to the oracle for a vertex, each of which is correct with probability 1/2 + ε. For this setting, we show an O(1)-approximation algorithm using O(n/ε²) total queries and Õ(m) runtime.

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Vladimir Braverman, Prathamesh Dharangutte, Vihan Shah, and Chen Wang. Learning-Augmented Maximum Independent Set. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) http://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.24

Author Details

Vladimir Braverman
  • Rice University, Houston, TX, USA
  • Google Research
Prathamesh Dharangutte
  • Rutgers University, NJ, USA
Vihan Shah
  • University of Waterloo, ON, Canada
Chen Wang
  • Rice University, Houston, TX, USA
  • Texas A&M University, College Station, TX, USA

Funding

  • Braverman, Vladimir: Supported partially by the Naval Research (ONR) grant N00014-23-1-2737, and NSF-CNS 2333887 award.
  • Dharangutte, Prathamesh: Supported by NSF through IIS-2229876 and CCF-2118953.
  • Shah, Vihan: Supported in part by Sepehr Assadi’s Sloan Research Fellowship and NSERC Discovery Grant.

Acknowledgements

The authors are grateful to Sepehr Assadi and Samson Zhou for the helpful conversations regarding the project. The authors also thank anonymous APPROX reviewers for helpful suggestions.

References

  1. Anders Aamand, Justin Y. Chen, Huy Lê Nguyen, Sandeep Silwal, and Ali Vakilian. Improved frequency estimation algorithms with and without predictions. CoRR, abs/2312.07535, 2023. URL: http://doi.org/10.48550/arXiv.2312.07535.
  2. Sungsoo Ahn, Younggyo Seo, and Jinwoo Shin. Learning what to defer for maximum independent sets. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pages 134-144. PMLR, 2020. URL: http://proceedings.mlr.press/v119/ahn20a.html.
  3. Vladimir E. Alekseev, Vadim V. Lozin, Dmitriy S. Malyshev, and Martin Milanic. The maximum independent set problem in planar graphs. In Edward Ochmanski and Jerzy Tyszkiewicz, editors, Mathematical Foundations of Computer Science 2008, 33rd International Symposium, MFCS 2008, Torun, Poland, August 25-29, 2008, Proceedings, volume 5162 of Lecture Notes in Computer Science, pages 96-107. Springer, 2008. URL: http://doi.org/10.1007/978-3-540-85238-4_7.
  4. Diogo V Andrade, Mauricio GC Resende, and Renato F Werneck. Fast local search for the maximum independent set problem. Journal of Heuristics, 18:525-547, 2012. Google Scholar
  5. Antonios Antoniadis, Hajo Broersma, and Yang Meng. Online graph coloring with predictions. arXiv preprint, 2023. URL: http://arxiv.org/abs/2312.00601.
  6. Yossi Azar, Debmalya Panigrahi, and Noam Touitou. Discrete-smoothness in online algorithms with predictions. Advances in Neural Information Processing Systems, 36, 2024. Google Scholar
  7. Eric Balkanski, Vasilis Gkatzelis, Xizhi Tan, and Cherlin Zhu. Online mechanism design with predictions. CoRR, abs/2310.02879, 2023. URL: http://doi.org/10.48550/arXiv.2310.02879.
  8. Etienne Bamas, Andreas Maggiori, and Ola Svensson. The primal-dual method for learning augmented algorithms. Advances in Neural Information Processing Systems, 33:20083-20094, 2020. Google Scholar
  9. Siddhartha Banerjee, Vincent Cohen-Addad, Anupam Gupta, and Zhouzi Li. Graph searching with predictions. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference, ITCS 2023, January 10-13, 2023, MIT, Cambridge, Massachusetts, USA, volume 251 of LIPIcs, pages 12:1-12:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: http://doi.org/10.4230/LIPICS.ITCS.2023.12.
  10. Ravi Boppana and Magnús M Halldórsson. Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2):180-196, 1992. Google Scholar
  11. Ravi B Boppana. Eigenvalues and graph bisection: An average-case analysis. In 28th Annual Symposium on Foundations of Computer Science (sfcs 1987), pages 280-285. IEEE, 1987. Google Scholar
  12. Nicolas Bourgeois, Bruno Escoffier, Vangelis Th. Paschos, and Johan M. M. van Rooij. Fast algorithms for max independent set. Algorithmica, 62(1-2):382-415, 2012. URL: http://doi.org/10.1007/S00453-010-9460-7.
  13. Jan van den Brand, Sebastian Forster, Yasamin Nazari, and Adam Polak. On dynamic graph algorithms with predictions. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3534-3557. SIAM, 2024. Google Scholar
  14. Lorenzo Brusca, Lars C. P. M. Quaedvlieg, Stratis Skoulakis, Grigorios Chrysos, and Volkan Cevher. Maximum independent set: Self-training through dynamic programming. In Alice Oh, Tristan Naumann, Amir Globerson, Kate Saenko, Moritz Hardt, and Sergey Levine, editors, Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans, LA, USA, December 10 - 16, 2023, 2023. URL: http://papers.nips.cc/paper_files/paper/2023/hash/7fe3170d88a8310ca86df2843f54236c-Abstract-Conference.html.
  15. Wei Cao, Jian Li, Yufei Tao, and Zhize Li. On top-k selection in multi-armed bandits and hidden bipartite graphs. In Corinna Cortes, Neil D. Lawrence, Daniel D. Lee, Masashi Sugiyama, and Roman Garnett, editors, Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, December 7-12, 2015, Montreal, Quebec, Canada, pages 1036-1044, 2015. URL: http://proceedings.neurips.cc/paper/2015/hash/ab233b682ec355648e7891e66c54191b-Abstract.html.
  16. Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In Claire Mathieu, editor, Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 892-901. SIAM, 2009. URL: http://doi.org/10.1137/1.9781611973068.97.
  17. Justin Chen, Sandeep Silwal, Ali Vakilian, and Fred Zhang. Faster fundamental graph algorithms via learned predictions. In International Conference on Machine Learning, pages 3583-3602. PMLR, 2022. Google Scholar
  18. Justin Y Chen, Talya Eden, Piotr Indyk, Honghao Lin, Shyam Narayanan, Ronitt Rubinfeld, Sandeep Silwal, Tal Wagner, David P Woodruff, and Michael Zhang. Triangle and four cycle counting with predictions in graph streams. arXiv preprint, 2022. URL: http://arxiv.org/abs/2203.09572.
  19. Lijie Chen, Jian Li, and Mingda Qiao. Nearly instance optimal sample complexity bounds for top-k arm selection. In Aarti Singh and Xiaojin (Jerry) Zhu, editors, Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, AISTATS 2017, 20-22 April 2017, Fort Lauderdale, FL, USA, volume 54 of Proceedings of Machine Learning Research, pages 101-110. PMLR, 2017. URL: http://proceedings.mlr.press/v54/chen17a.html.
  20. Jakub Chledowski, Adam Polak, Bartosz Szabucki, and Konrad Tomasz Zolna. Robust learning-augmented caching: An experimental study. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pages 1920-1930. PMLR, 2021. URL: http://proceedings.mlr.press/v139/chledowski21a.html.
  21. Nicolas Christianson, Tinashe Handina, and Adam Wierman. Chasing convex bodies and functions with black-box advice. In Po-Ling Loh and Maxim Raginsky, editors, Conference on Learning Theory, 2-5 July 2022, London, UK, volume 178 of Proceedings of Machine Learning Research, pages 867-908. PMLR, 2022. URL: http://proceedings.mlr.press/v178/christianson22a.html.
  22. Julia Chuzhoy and Alina Ene. On approximating maximum independent set of rectangles. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 820-829. IEEE Computer Society, 2016. URL: http://doi.org/10.1109/FOCS.2016.92.
  23. Vincent Cohen-Addad, Tommaso d'Orsi, Anupam Gupta, Euiwoong Lee, and Debmalya Panigrahi. Max-cut with ε-accurate predictions. CoRR, abs/2402.18263, 2024. URL: http://doi.org/10.48550/arXiv.2402.18263.
  24. Jakob Dahlum, Sebastian Lamm, Peter Sanders, Christian Schulz, Darren Strash, and Renato F. Werneck. Accelerating local search for the maximum independent set problem. In Andrew V. Goldberg and Alexander S. Kulikov, editors, Experimental Algorithms - 15th International Symposium, SEA 2016, St. Petersburg, Russia, June 5-8, 2016, Proceedings, volume 9685 of Lecture Notes in Computer Science, pages 118-133. Springer, 2016. URL: http://doi.org/10.1007/978-3-319-38851-9_9.
  25. Devdatt P Dubhashi and Alessandro Panconesi. Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, 2009. Google Scholar
  26. Jon C. Ergun, Zhili Feng, Sandeep Silwal, David P. Woodruff, and Samson Zhou. Learning-augmented dollarkdollar-means clustering. In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022. OpenReview.net, 2022. URL: http://openreview.net/forum?id=X8cLTHexYyY.
  27. Eyal Even-Dar, Shie Mannor, and Yishay Mansour. PAC bounds for multi-armed bandit and markov decision processes. In Jyrki Kivinen and Robert H. Sloan, editors, Computational Learning Theory, 15th Annual Conference on Computational Learning Theory, COLT 2002, Sydney, Australia, July 8-10, 2002, Proceedings, volume 2375 of Lecture Notes in Computer Science, pages 255-270. Springer, 2002. URL: http://doi.org/10.1007/3-540-45435-7_18.
  28. Eyal Even-Dar, Shie Mannor, and Yishay Mansour. Action elimination and stopping conditions for the multi-armed bandit and reinforcement learning problems. J. Mach. Learn. Res., 7:1079-1105, 2006. URL: http://jmlr.org/papers/v7/evendar06a.html.
  29. Uriel Feige. Approximating maximum clique by removing subgraphs. SIAM Journal on Discrete Mathematics, 18(2):219-225, 2004. Google Scholar
  30. Willem Feijen and Guido Schäfer. Using machine learning predictions to speed-up dijkstra’s shortest path algorithm. CoRR, pages 1-28, 2021. Google Scholar
  31. Fedor V. Fomin, Fabrizio Grandoni, and Dieter Kratsch. Measure and conquer: a simple o(2^0.288n) independent set algorithm. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 18-25. ACM Press, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109560.
  32. Waldo Gálvez, Arindam Khan, Mathieu Mari, Tobias Mömke, Madhusudhan Reddy Pittu, and Andreas Wiese. A 3-approximation algorithm for maximum independent set of rectangles. 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 894-905. SIAM, 2022. URL: http://doi.org/10.1137/1.9781611977073.38.
  33. Suprovat Ghoshal, Konstantin Makarychev, and Yury Makarychev. Constraint satisfaction problems with advice. arXiv preprint, 2024. URL: http://arxiv.org/abs/2403.02212.
  34. Francesco Gullo, Domenico Mandaglio, and Andrea Tagarelli. A combinatorial multi-armed bandit approach to correlation clustering. Data Min. Knowl. Discov., 37(4):1630-1691, 2023. URL: http://doi.org/10.1007/S10618-023-00937-5.
  35. Shubham Gupta, Peter W. J. Staar, and Christian de Sainte Marie. Clustering items from adaptively collected inconsistent feedback. In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li, editors, International Conference on Artificial Intelligence and Statistics, 2-4 May 2024, Palau de Congressos, Valencia, Spain, volume 238 of Proceedings of Machine Learning Research, pages 604-612. PMLR, 2024. URL: http://proceedings.mlr.press/v238/gupta24a.html.
  36. Johan Håstad. Clique is hard to approximate within n^(1-ε). In 37th Annual Symposium on Foundations of Computer Science, FOCS '96, Burlington, Vermont, USA, 14-16 October, 1996, pages 627-636. IEEE Computer Society, 1996. URL: http://doi.org/10.1109/SFCS.1996.548522.
  37. Monika Henzinger, Andrea Lincoln, Barna Saha, Martin P Seybold, and Christopher Ye. On the complexity of algorithms with predictions for dynamic graph problems. arXiv preprint, 2023. URL: http://arxiv.org/abs/2307.16771.
  38. Chen-Yu Hsu, Piotr Indyk, Dina Katabi, and Ali Vakilian. Learning-based frequency estimation algorithms. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019. URL: http://openreview.net/forum?id=r1lohoCqY7.
  39. Bingbing Hu, Evangelos Kosinas, and Adam Polak. Connectivity oracles for predictable vertex failures. arXiv preprint, 2023. URL: http://arxiv.org/abs/2312.08489.
  40. Shivaram Kalyanakrishnan and Peter Stone. Efficient selection of multiple bandit arms: Theory and practice. In Johannes Fürnkranz and Thorsten Joachims, editors, Proceedings of the 27th International Conference on Machine Learning (ICML-10), June 21-24, 2010, Haifa, Israel, pages 511-518. Omnipress, 2010. URL: http://icml.cc/Conferences/2010/papers/410.pdf.
  41. Shivaram Kalyanakrishnan, Ambuj Tewari, Peter Auer, and Peter Stone. PAC subset selection in stochastic multi-armed bandits. In Proceedings of the 29th International Conference on Machine Learning, ICML 2012, Edinburgh, Scotland, UK, June 26 - July 1, 2012. icml.cc / Omnipress, 2012. URL: http://icml.cc/2012/papers/359.pdf.
  42. Richard M Karp. Reducibility among combinatorial problems. Springer, 2010. Google Scholar
  43. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapproximability results for max-cut and other 2-variable csps? SIAM Journal on Computing, 37(1):319-357, 2007. Google Scholar
  44. Yuko Kuroki, Atsushi Miyauchi, Francesco Bonchi, and Wei Chen. Query-efficient correlation clustering with noisy oracle. CoRR, abs/2402.01400, 2024. URL: http://doi.org/10.48550/arXiv.2402.01400.
  45. Silvio Lattanzi, Ola Svensson, and Sergei Vassilvitskii. Speeding up bellman ford via minimum violation permutations. In International Conference on Machine Learning, pages 18584-18598. PMLR, 2023. Google Scholar
  46. Thomas Lavastida, Benjamin Moseley, R. Ravi, and Chenyang Xu. Learnable and instance-robust predictions for online matching, flows and load balancing. 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 59:1-59:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: http://doi.org/10.4230/LIPICS.ESA.2021.59.
  47. Avner Magen and Mohammad Moharrami. Robust algorithms for MAX INDEPENDENT SET on minor-free graphs based on the sherali-adams hierarchy. In Irit Dinur, Klaus Jansen, Joseph Naor, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 12th International Workshop, APPROX 2009, and 13th International Workshop, RANDOM 2009, Berkeley, CA, USA, August 21-23, 2009. Proceedings, volume 5687 of Lecture Notes in Computer Science, pages 258-271. Springer, 2009. URL: http://doi.org/10.1007/978-3-642-03685-9_20.
  48. Michael Mitzenmacher and Sergei Vassilvitskii. Algorithms with predictions. Communications of the ACM, 65(7):33-35, 2022. Google Scholar
  49. Thomas Pontoizeau, Florian Sikora, Florian Yger, and Tristan Cazenave. Neural maximum independent set. In Machine Learning and Principles and Practice of Knowledge Discovery in Databases - International Workshops of ECML PKDD 2021, Virtual Event, September 13-17, 2021, Proceedings, Part I, volume 1524 of Communications in Computer and Information Science, pages 223-237. Springer, 2021. URL: http://doi.org/10.1007/978-3-030-93736-2_18.
  50. Manish Purohit, Zoya Svitkina, and Ravi Kumar. Improving online algorithms via ML predictions. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett, editors, Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, pages 9684-9693, 2018. URL: http://proceedings.neurips.cc/paper/2018/hash/73a427badebe0e32caa2e1fc7530b7f3-Abstract.html.
  51. John M Robson. Finding a maximum independent set in time o (2n/4). Technical report, Technical Report 1251-01, LaBRI, Université Bordeaux I, 2001. Google Scholar
  52. Tim Roughgarden. Beyond the worst-case analysis of algorithms. Cambridge University Press, 2021. Google Scholar
  53. Karim Abdel Sadek and Marek Elias. Algorithms for caching and mts with reduced number of predictions. arXiv preprint, 2024. URL: http://arxiv.org/abs/2404.06280.
  54. Jeanette P. Schmidt, Alan Siegel, and Aravind Srinivasan. Chernoff-hoeffding bounds for applications with limited independence. SIAM J. Discret. Math., 8(2):223-250, 1995. Google Scholar
  55. Max Simchowitz, Kevin G. Jamieson, and Benjamin Recht. The simulator: Understanding adaptive sampling in the moderate-confidence regime. In Satyen Kale and Ohad Shamir, editors, Proceedings of the 30th Conference on Learning Theory, COLT 2017, Amsterdam, The Netherlands, 7-10 July 2017, volume 65 of Proceedings of Machine Learning Research, pages 1794-1834. PMLR, 2017. URL: http://proceedings.mlr.press/v65/simchowitz17a.html.
  56. Clifford Stein and Hao-Ting Wei. Learning-augmented online packet scheduling with deadlines. CoRR, abs/2305.07164, 2023. URL: http://doi.org/10.48550/arXiv.2305.07164.
  57. Jose L. Walteros and Austin Buchanan. Why is maximum clique often easy in practice? Oper. Res., 68(6):1866-1895, 2020. URL: http://doi.org/10.1287/OPRE.2019.1970.
  58. Jinghui Xia and Zengfeng Huang. Optimal clustering with noisy queries via multi-armed bandit. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvári, Gang Niu, and Sivan Sabato, editors, International Conference on Machine Learning, ICML 2022, 17-23 July 2022, Baltimore, Maryland, USA, volume 162 of Proceedings of Machine Learning Research, pages 24315-24331. PMLR, 2022. URL: http://proceedings.mlr.press/v162/xia22a.html.
  59. Mingyu Xiao and Hiroshi Nagamochi. Exact algorithms for maximum independent set. In Leizhen Cai, Siu-Wing Cheng, and Tak Wah Lam, editors, Algorithms and Computation - 24th International Symposium, ISAAC 2013, Hong Kong, China, December 16-18, 2013, Proceedings, volume 8283 of Lecture Notes in Computer Science, pages 328-338. Springer, 2013. URL: http://doi.org/10.1007/978-3-642-45030-3_31.
  60. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 681-690, 2006. Google Scholar
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