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Fault Tolerant Max-Cut

Authors Keren Censor-Hillel, Noa Marelly, Roy Schwartz, Tigran Tonoyan

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Author Details

Keren Censor-Hillel
  • Department of Computer Science, Technion, Haifa, Israel
Noa Marelly
  • Department of Computer Science, Technion, Haifa, Israel
Roy Schwartz
  • Department of Computer Science, Technion, Haifa, Israel
Tigran Tonoyan
  • Department of Computer Science, Technion, Haifa, Israel

Funding

This project has received funding from the European Research Council (ERC), under the European Unions Horizon 2020 research and innovation programme under grant agreement No 755839, and from Israel Science Foundation (ISF), under grant No 1336/16.

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Keren Censor-Hillel, Noa Marelly, Roy Schwartz, and Tigran Tonoyan. Fault Tolerant Max-Cut. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.46

Abstract

In this work, we initiate the study of fault tolerant Max-Cut, where given an edge-weighted undirected graph G = (V,E), the goal is to find a cut S ⊆ V that maximizes the total weight of edges that cross S even after an adversary removes k vertices from G. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures k we present an approximation of (0.878-ε) against an adaptive adversary and of α_{GW}≈ 0.8786 against an oblivious adversary (here α_{GW} is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM `95]). Additionally, we present a hardness of approximation of α_{GW} against both types of adversaries, rendering our results (virtually) tight. The non-linear nature of the fault tolerant objective makes the design and analysis of algorithms harder when compared to the classic Max-Cut. Hence, we employ approaches ranging from multi-objective optimization to LP duality and the ellipsoid algorithm to obtain our results.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • fault-tolerance
  • max-cut
  • approximation

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References

  1. Noga Alon, Shiri Chechik, and Sarel Cohen. Deterministic combinatorial replacement paths and distance sensitivity oracles. 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 12:1-12:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.12.
  2. Noga Alon and Assaf Naor. Approximating the cut-norm via grothendieck’s inequality. SIAM J. Comput., 35(4):787-803, 2006. Google Scholar
  3. Eric Angel, Evripidis Bampis, and Laurent Gourvès. Approximation algorithms for the bi-criteria weighted MAX-CUT problem. Discret. Appl. Math., 154(12):1685-1692, 2006. URL: https://doi.org/10.1016/j.dam.2006.02.008.
  4. Takao Asano and David P Williamson. Improved approximation algorithms for MAX SAT. Journal of Algorithms, 42(1):173-202, 2002. Google Scholar
  5. Per Austrin, Siavosh Benabbas, and Konstantinos Georgiou. Better balance by being biased: A 0.8776-approximation for Max Bisection. ACM Trans. Algorithms, 13(1):2:1-2:27, 2016. Google Scholar
  6. Dmitrii Avdiukhin, Slobodan Mitrovic, Grigory Yaroslavtsev, and Samson Zhou. Adversarially robust submodular maximization under knapsack constraints. In Ankur Teredesai, Vipin Kumar, Ying Li, Rómer Rosales, Evimaria Terzi, and George Karypis, editors, Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD 2019, Anchorage, AK, USA, August 4-8, 2019, pages 148-156. ACM, 2019. URL: https://doi.org/10.1145/3292500.3330911.
  7. Adi Avidor, Ido Berkovitch, and Uri Zwick. Improved approximation algorithms for MAX NAE-SAT and MAX SAT. In Thomas Erlebach and Giuseppe Persiano, editors, Approximation and Online Algorithms, Third International Workshop, WAOA 2005, Palma de Mallorca, Spain, October 6-7, 2005, Revised Papers, volume 3879 of Lecture Notes in Computer Science, pages 27-40. Springer, 2005. URL: https://doi.org/10.1007/11671411_3.
  8. Surender Baswana, Keerti Choudhary, Moazzam Hussain, and Liam Roditty. Approximate single source fault tolerant shortest path. 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 1901-1915. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.124.
  9. Surender Baswana, Keerti Choudhary, and Liam Roditty. Fault tolerant reachability for directed graphs. In Yoram Moses, editor, Distributed Computing - 29th International Symposium, DISC 2015, Tokyo, Japan, October 7-9, 2015, Proceedings, volume 9363 of Lecture Notes in Computer Science, pages 528-543. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48653-5_35.
  10. Surender Baswana, Keerti Choudhary, and Liam Roditty. Fault tolerant subgraph for single source reachability: generic and optimal. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 509-518. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897648.
  11. Amey Bhangale and Subhash Khot. Simultaneous max-cut is harder to approximate than max-cut. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 9:1-9:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.9.
  12. Amey Bhangale, Subhash Khot, Swastik Kopparty, Sushant Sachdeva, and Devanathan Thiruvenkatachari. Near-optimal approximation algorithm for simultaneous max-cut. 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 1407-1425. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.93.
  13. Amey Bhangale, Swastik Kopparty, and Sushant Sachdeva. Simultaneous approximation of constraint satisfaction problems. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 193-205. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_16.
  14. Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Fault-tolerant approximate shortest-path trees. Algorithmica, 80(12):3437-3460, 2018. URL: https://doi.org/10.1007/s00453-017-0396-z.
  15. Greg Bodwin, Michael Dinitz, Merav Parter, and Virginia Vassilevska Williams. Optimal vertex fault tolerant spanners (for fixed stretch). 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 1884-1900. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.123.
  16. Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams. Preserving distances in very faulty graphs. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 73:1-73:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.73.
  17. Greg Bodwin and Shyamal Patel. A trivial yet optimal solution to vertex fault tolerant spanners. In Peter Robinson and Faith Ellen, editors, Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC 2019, Toronto, ON, Canada, July 29 - August 2, 2019, pages 541-543. ACM, 2019. URL: https://doi.org/10.1145/3293611.3331588.
  18. Austin Buchanan, Je Sang Sung, Sergiy Butenko, and Eduardo L. Pasiliao. An integer programming approach for fault-tolerant connected dominating sets. INFORMS J. Comput., 27(1):178-188, 2015. URL: https://doi.org/10.1287/ijoc.2014.0619.
  19. Keren Censor-Hillel, Noa Marelly, Roy Schwartz, and Tigran Tonoyan. Fault tolerant max-cut. CoRR, arXiv:2105.01138, 2021. URL: http://arxiv.org/abs/2105.01138.
  20. Moses Charikar, Venkatesan Guruswami, and Anthony Wirth. Clustering with qualitative information. J. Comput. Syst. Sci., 71(3):360-383, 2005. Google Scholar
  21. Shiri Chechik and Sarel Cohen. Near optimal algorithms for the single source replacement paths problem. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 2090-2109. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.126.
  22. Shiri Chechik and Ofer Magen. Near optimal algorithm for the directed single source replacement paths problem. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 81:1-81:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.81.
  23. Shiri Chechik and David Peleg. Robust fault tolerant uncapacitated facility location. Theor. Comput. Sci., 543:9-23, 2014. URL: https://doi.org/10.1016/j.tcs.2014.05.013.
  24. Michael Dinitz and Robert Krauthgamer. Directed spanners via flow-based linear programs. In Lance Fortnow and Salil P. Vadhan, editors, Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 323-332. ACM, 2011. URL: https://doi.org/10.1145/1993636.1993680.
  25. Michael Dinitz and Robert Krauthgamer. Fault-tolerant spanners: better and simpler. In Cyril Gavoille and Pierre Fraigniaud, editors, Proceedings of the 30th Annual ACM Symposium on Principles of Distributed Computing, PODC 2011, San Jose, CA, USA, June 6-8, 2011, pages 169-178. ACM, 2011. URL: https://doi.org/10.1145/1993806.1993830.
  26. Uriel Feige and Michel Goemans. Approximating the value of two power proof systems, with applications to Max 2-SAT and Max DiCut. In istcs, page 0182. IEEE, 1995. Google Scholar
  27. M. R. Garey, David S. Johnson, and Larry J. Stockmeyer. Some simplified np-complete graph problems. Theor. Comput. Sci., 1(3):237-267, 1976. URL: https://doi.org/10.1016/0304-3975(76)90059-1.
  28. Mohsen Ghaffari and Merav Parter. Near-optimal distributed algorithms for fault-tolerant tree structures. In Christian Scheideler and Seth Gilbert, editors, Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2016, Asilomar State Beach/Pacific Grove, CA, USA, July 11-13, 2016, pages 387-396. ACM, 2016. URL: https://doi.org/10.1145/2935764.2935795.
  29. 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.
  30. Fabrizio Grandoni and Virginia Vassilevska Williams. Improved distance sensitivity oracles via fast single-source replacement paths. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 748-757. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/FOCS.2012.17.
  31. Martin Grötschel, László Lovász, and Alexander Schrijver. The Ellipsoid Method, pages 64-101. Springer Berlin Heidelberg, Berlin, Heidelberg, 1993. URL: https://doi.org/10.1007/978-3-642-78240-4_4.
  32. Sudipto Guha, Adam Meyerson, and Kamesh Munagala. A constant factor approximation algorithm for the fault-tolerant facility location problem. J. Algorithms, 48(2):429-440, 2003. URL: https://doi.org/10.1016/S0196-6774(03)00056-7.
  33. Manoj Gupta and Shahbaz Khan. Multiple source dual fault tolerant BFS trees. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 127:1-127:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.127.
  34. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
  35. Kamal Jain, Mohammad Mahdian, and Mohammad R. Salavatipour. Packing steiner trees. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA, pages 266-274. ACM/SIAM, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644154.
  36. Kamal Jain and Vijay V. Vazirani. An approximation algorithm for the fault tolerant metric facility location problem. Algorithmica, 38(3):433-439, 2004. URL: https://doi.org/10.1007/s00453-003-1070-1.
  37. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  38. 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.
  39. 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.
  40. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. J. Comput. Syst. Sci., 74(3):335-349, 2008. URL: https://doi.org/10.1016/j.jcss.2007.06.019.
  41. Christos Levcopoulos, Giri Narasimhan, and Michiel H. M. Smid. Improved algorithms for constructing fault-tolerant spanners. Algorithmica, 32(1):144-156, 2002. URL: https://doi.org/10.1007/s00453-001-0075-x.
  42. Michael Lewin, Dror Livnat, and Uri Zwick. Improved rounding techniques for the MAX 2-sat and MAX DI-CUT problems. In William J. Cook and Andreas S. Schulz, editors, Integer Programming and Combinatorial Optimization, 9th International IPCO Conference, Cambridge, MA, USA, May 27-29, 2002, Proceedings, volume 2337 of Lecture Notes in Computer Science, pages 67-82. Springer, 2002. URL: https://doi.org/10.1007/3-540-47867-1_6.
  43. Sanjeev Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM J. Comput., 28(5):1641-1663, 1999. URL: https://doi.org/10.1137/S0097539796309326.
  44. Shiro Matuura and Tomomi Matsui. 63-approximation algorithm for MAX DICUT. In Michel X. Goemans, Klaus Jansen, José D. P. Rolim, and Luca Trevisan, editors, Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques, 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001 Berkeley, CA, USA, August 18-20, 2001, Proceedings, volume 2129 of Lecture Notes in Computer Science, pages 138-146. Springer, 2001. URL: https://doi.org/10.1007/3-540-44666-4_17.
  45. Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. In Foundations of Computer Science, (FOCS), pages 21-30, 2005. Google Scholar
  46. James B. Orlin, Andreas S. Schulz, and Rajan Udwani. Robust monotone submodular function maximization. In Quentin Louveaux and Martin Skutella, editors, Integer Programming and Combinatorial Optimization - 18th International Conference, IPCO 2016, Liège, Belgium, June 1-3, 2016, Proceedings, volume 9682 of Lecture Notes in Computer Science, pages 312-324. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-33461-5_26.
  47. Merav Parter. Vertex fault tolerant additive spanners. In Fabian Kuhn, editor, Distributed Computing - 28th International Symposium, DISC 2014, Austin, TX, USA, October 12-15, 2014. Proceedings, volume 8784 of Lecture Notes in Computer Science, pages 167-181. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-45174-8_12.
  48. Merav Parter. Dual failure resilient BFS structure. In Chryssis Georgiou and Paul G. Spirakis, editors, Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015, Donostia-San Sebastián, Spain, July 21 - 23, 2015, pages 481-490. ACM, 2015. URL: https://doi.org/10.1145/2767386.2767408.
  49. Merav Parter. Distributed constructions of dual-failure fault-tolerant distance preservers. In Hagit Attiya, editor, 34th International Symposium on Distributed Computing, DISC 2020, October 12-16, 2020, Virtual Conference, volume 179 of LIPIcs, pages 21:1-21:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.21.
  50. Merav Parter and David Peleg. Sparse fault-tolerant BFS trees. CoRR, abs/1302.5401, 2013. URL: http://arxiv.org/abs/1302.5401.
  51. Merav Parter and David Peleg. Fault tolerant approximate BFS structures. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1073-1092. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.80.
  52. Merav Parter and David Peleg. Fault tolerant BFS structures: A reinforcement-backup tradeoff. In Guy E. Blelloch and Kunal Agrawal, editors, Proceedings of the 27th ACM on Symposium on Parallelism in Algorithms and Architectures, SPAA 2015, Portland, OR, USA, June 13-15, 2015, pages 264-273. ACM, 2015. URL: https://doi.org/10.1145/2755573.2755590.
  53. Prasad Raghavendra and Ning Tan. Approximating csps with global cardinality constraints using SDP hierarchies. In Symposium on Discrete Algorithms (SODA), pages 373-387. SIAM, 2012. Google Scholar
  54. Liam Roditty and Uri Zwick. Replacement paths and k simple shortest paths in unweighted directed graphs. ACM Trans. Algorithms, 8(4):33:1-33:11, 2012. URL: https://doi.org/10.1145/2344422.2344423.
  55. Shay Solomon. From hierarchical partitions to hierarchical covers: optimal fault-tolerant spanners for doubling metrics. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 363-372. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591864.
  56. Chaitanya Swamy. Correlation clustering: Maximizing agreements via semidefinite programming. In Symposium on Discrete Algorithms, SODA, pages 526-527, 2004. Google Scholar
  57. Chaitanya Swamy and David B. Shmoys. Fault-tolerant facility location. ACM Trans. Algorithms, 4(4):51:1-51:27, 2008. URL: https://doi.org/10.1145/1383369.1383382.
  58. Luca Trevisan, Gregory B. Sorkin, Madhu Sudan, and David P. Williamson. Gadgets, approximation, and linear programming. SIAM J. Comput., 29(6):2074-2097, 2000. URL: https://doi.org/10.1137/S0097539797328847.
  59. Jiao Zhou, Zhao Zhang, Shaojie Tang, Xiaohui Huang, and Ding-Zhu Du. Breaking the O(ln n) barrier: An enhanced approximation algorithm for fault-tolerant minimum weight connected dominating set. INFORMS J. Comput., 30(2):225-235, 2018. URL: https://doi.org/10.1287/ijoc.2017.0775.
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