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Fast Distributed Approximation for Max-Cut

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Algorithms for Sensor Systems (ALGOSENSORS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 10718))

Abstract

Finding a maximum cut is a fundamental task in many computational settings, with a central application in wireless networks. Surprisingly, Max-Cut has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the \(\mathcal {LOCAL}\) or \(\mathcal {CONGEST}\) models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any \(\epsilon > 0\), we develop randomized approximation algorithms achieving a ratio of \((1-\varepsilon )\) to the optimum for Max-Cut on bipartite graphs in the \(\mathcal {CONGEST}\) model, and on general graphs in the \(\mathcal {LOCAL}\) model.

We further present efficient deterministic algorithms, including a 1/3-approximation for Max-Dicut in our models, thus improving the best known (randomized) ratio of 1/4. Our algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal Computing 44:1384–1402, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest.

K. Censor-Hillel—The research is supported in part by the Israel Science Foundation (grant 1696/14).

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Notes

  1. 1.

    Max-Cut naturally arises also in VLSI [9], statistical physics [4] and machine learning [47].

  2. 2.

    In Max-Dicut we seek the maximum size edge-set crossing from S to \(\bar{S}\).

  3. 3.

    Due to space constraints, some of the results are omitted. A detailed version of this paper can be found in [8].

  4. 4.

    This assumption is needed only for the \((\varDelta +1)\)-coloring algorithm [6] used in Sect. 4; it can be omitted (see [6]), increasing the running time by a constant factor.

  5. 5.

    Our algorithm can be viewed as one phase of the distributed algorithm presented by Elkin et al. in [12] with some necessary changes.

  6. 6.

    This can be done by running a BFS in parallel from all vertices. Each vertex propagates the information from the root with lowest id it knows so far, and joins its tree. Thus, at the end of the process, we have a BFS tree rooted at the vertex with the lowest id.

References

  1. Åstrand, M., Floréen, P., Polishchuk, V., Rybicki, J., Suomela, J., Uitto, J.: A local 2-approximation algorithm for the vertex cover problem. In: Keidar, I. (ed.) DISC 2009. LNCS, vol. 5805, pp. 191–205. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04355-0_21

    Chapter  Google Scholar 

  2. Åstrand, M., Suomela, J.: Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks. In: Proceedings of the Twenty-Second Annual ACM Symposium on Parallelism in Algorithms and Architectures, pp. 294–302. ACM (2010)

    Google Scholar 

  3. Bar-Yehuda, R., Censor-Hillel, K., Schwartzman, G.: A distributed (2+\(\epsilon \))-approximation for vertex cover in O(log\(\varDelta \)/\(\epsilon \) log log \(\varDelta \)) rounds. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, 25–28 July 2016, pp. 3–8 (2016)

    Google Scholar 

  4. Barahona, F., Grötschel, M., Jünger, M., Reinelt, G.: An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res. 36(3), 493–513 (1988)

    Article  MATH  Google Scholar 

  5. da Ponte Barbosa, R., Ene, A., Nguyen, H.L., Ward, J.: A new framework for distributed submodular maximization. arXiv preprint http://arxiv.org/abs/1507.03719 (2015)

  6. Barenboim, L.: Deterministic (\(\delta \)+ 1)-coloring in sublinear (in \(\delta \)) time in static, dynamic and faulty networks. In: Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pp. 345–354. ACM (2015)

    Google Scholar 

  7. Buchbinder, N., Feldman, M., Naor, J., Schwartz, R.: A tight linear time (1/2)-approximation for unconstrained submodular maximization. SIAM J. Comput. 44(5), 1384–1402 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Censor-Hillel, K., Levy, R., Shachnai, H.: Fast distributed approximation for max-cut. arXiv preprint http://arxiv.org/abs/1707.08496 (2017)

  9. Chang, K., Du, D.C.: Efficient algorithms for layer assignment problem. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 6(1), 67–78 (1987)

    Article  Google Scholar 

  10. Chin, K.W., Soh, S., Meng, C.: Novel scheduling algorithms for concurrent transmit/receive wireless mesh networks. Comput. Netw. 56(4), 1200–1214 (2012)

    Article  Google Scholar 

  11. Elkin, M.: Distributed approximation: a survey. ACM SIGACT News 35(4), 40–57 (2004)

    Article  Google Scholar 

  12. Elkin, M., Neiman, O.: Distributed strong diameter network decomposition. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, pp. 211–216. ACM (2016)

    Google Scholar 

  13. Elkin, M., Neiman, O.: Efficient algorithms for constructing very sparse spanners and emulators. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, 16–19 January, pp. 652–669 (2017)

    Google Scholar 

  14. Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. SIAM J. Comput. 40(4), 1133–1153 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1(3), 237–267 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghaffari, M., Kuhn, F.: Distributed minimum cut approximation. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 1–15. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41527-2_1

    Chapter  Google Scholar 

  17. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grötschel, M., Pulleyblank, W.R.: Weakly bipartite graphs and the max-cut problem. Oper. Res. Lett. 1(1), 23–27 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hadlock, F.: Finding a maximum cut of a planar graph in polynomial time. SIAM J. Comput. 4(3), 221–225 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  20. Håstad, J.: Some optimal inapproximability results. J. ACM (JACM) 48(4), 798–859 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Henzinger, M., Krinninger, S., Nanongkai, D.: A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pp. 489–498. ACM (2016)

    Google Scholar 

  22. Hirvonen, J., Rybicki, J., Schmid, S., Suomela, J.: Large cuts with local algorithms on triangle-free graphs. arXiv preprint arXiv:1402.2543 (2014)

  23. Kale, S., Seshadhri, C.: Combinatorial approximation algorithms for maxcut using random walks. arXiv preprint arXiv:1008.3938 (2010)

  24. Kapralov, M., Khanna, S., Sudan, M.: Streaming lower bounds for approximating max-cut. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1263–1282. SIAM (2015)

    Google Scholar 

  25. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, Heidelberg (1972). https://doi.org/10.1007/978-1-4684-2001-2_9

  26. Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproximability results for max-cut and other 2-variable CSPs? SIAM J. Comput. 37(1), 319–357 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Komurlu, C., Bilgic, M.: Active inference and dynamic Gaussian Bayesian networks for battery optimization in wireless sensor networks. In: AI for Smart Grids and Smart Buildings, Papers from the 2016 AAAI Workshop, Phoenix, Arizona, USA (2016)

    Google Scholar 

  28. Kuhn, F., Moscibroda, T.: Distributed approximation of capacitated dominating sets. Theory Comput. Syst. 47(4), 811–836 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kuhn, F., Moscibroda, T., Wattenhofer, R.: Local computation: lower and upper bounds. J. ACM (JACM) 63(2), 17 (2016)

    Article  MathSciNet  Google Scholar 

  30. Lenzen, C., Pignolet, Y.A., Wattenhofer, R.: Distributed minimum dominating set approximations in restricted families of graphs. Distrib. Comput. 26(2), 119–137 (2013)

    Article  MATH  Google Scholar 

  31. Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lotker, Z., Patt-Shamir, B., Pettie, S.: Improved distributed approximate matching. In: Proceedings of the Twentieth Annual Symposium on Parallelism in Algorithms and Architectures, pp. 129–136. ACM (2008)

    Google Scholar 

  33. Matuura, S., Matsui, T.: 0.863-approximation algorithm for MAX DICUT. In: Goemans, M., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX/RANDOM -2001. LNCS, vol. 2129, pp. 138–146. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44666-4_17

    Chapter  Google Scholar 

  34. Miller, G.L., Peng, R., Xu, S.C.: Parallel graph decompositions using random shifts. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Parallelism in Algorithms and Architectures, pp. 196–203. ACM (2013)

    Google Scholar 

  35. Mirrokni, V., Zadimoghaddam, M.: Randomized composable core-sets for distributed submodular maximization. In: Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pp. 153–162. ACM (2015)

    Google Scholar 

  36. Mirzasoleiman, B., Karbasi, A., Sarkar, R., Krause, A.: Distributed submodular maximization: identifying representative elements in massive data. In: Advances in Neural Information Processing Systems, pp. 2049–2057 (2013)

    Google Scholar 

  37. Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  38. Motwani, R., Raghavan, P.: Randomized Algorithms. Chapman & Hall/CRC, London (2010)

    Google Scholar 

  39. Nanongkai, D.: Distributed approximation algorithms for weighted shortest paths. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp. 565–573. ACM (2014)

    Google Scholar 

  40. Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 229–234. ACM (1988)

    Google Scholar 

  41. Peleg, D.: Distributed Computing. SIAM Monographs on Discrete Mathematics and Applications, vol. 5 (2000)

    Google Scholar 

  42. Sahni, S., Gonzalez, T.: P-complete approximation problems. J. ACM (JACM) 23(3), 555–565 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  43. Saurabh, S., Zehavi, M.: \((k,n-k)\)-Max-Cut: an \({\cal{O}}^*(2^p)\)-time algorithm and a polynomial kernel. In: Kranakis, E., Navarro, G., Chávez, E. (eds.) LATIN 2016. LNCS, vol. 9644, pp. 686–699. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49529-2_51

    Chapter  Google Scholar 

  44. Tangwongsan, K.: Efficient parallel approximation algorithms. Ph.D. thesis, School of Computer Science, Carnegie Mellon University (2011)

    Google Scholar 

  45. Trevisan, L.: Max cut and the smallest eigenvalue. SIAM J. Comput. 41(6), 1769–1786 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Trevisan, L., Sorkin, G.B., Sudan, M., Williamson, D.P.: Gadgets, approximation, and linear programming. SIAM J. Comput. 29(6), 2074–2097 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  47. Wang, J., Jebara, T., Chang, S.F.: Semi-supervised learning using greedy max-cut. J. Mach. Learn. Res. 14(Mar), 771–800 (2013)

    Google Scholar 

  48. Wang, L., Chin, K., Soh, S.: Joint routing and scheduling in multi-Tx/Rx wireless mesh networks with random demands. Comput. Netw. 98, 44–56 (2016)

    Article  Google Scholar 

  49. Wang, W., Liu, B., Yang, M., Luo, J., Shen, X.: Max-cut based overlapping channel assignment for 802.11 multi-radio wireless mesh networks. In: 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design (CSCWD), pp. 662–667 (2013)

    Google Scholar 

  50. Xu, Y., Chin, K., Raad, R., Soh, S.: A novel distributed max-weight link scheduler for multi-transmit/receive wireless mesh networks. IEEE Trans. Veh. Technol. 65(11), 9345–9357 (2016)

    Article  Google Scholar 

  51. Xue, G., He, Q., Zhu, H., He, T., Liu, Y.: Sociality-aware access point selection in enterprise wireless LANs. IEEE Trans. Parallel Distrib. Syst. 24(10), 2069–2078 (2013)

    Article  Google Scholar 

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Acknowledgements

We thank Roy Schwartz and Shay Kutten for stimulating discussions and for helpful comments on the paper.

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Authors and Affiliations

  1. Computer Science Department, Technion, 3200003, Haifa, Israel

    Keren Censor-Hillel, Rina Levy & Hadas Shachnai

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  1. Keren Censor-Hillel
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Correspondence to Keren Censor-Hillel .

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Editors and Affiliations

  1. IMDEA Networks Institute, Leganés, Spain

    Antonio Fernández Anta

  2. University of Wrocław, Wroclaw, Poland

    Tomasz Jurdzinski

  3. Pace University, New York, USA

    Miguel A. Mosteiro

  4. Rutgers University, North Brunswick, New Jersey, USA

    Yanyong Zhang

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Censor-Hillel, K., Levy, R., Shachnai, H. (2017). Fast Distributed Approximation for Max-Cut. In: Fernández Anta, A., Jurdzinski, T., Mosteiro, M., Zhang, Y. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2017. Lecture Notes in Computer Science(), vol 10718. Springer, Cham. https://doi.org/10.1007/978-3-319-72751-6_4

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