On multiplicative weight updates for concave and submodular function maximization [extended abstract]

ITCS 2015 - Proceedings of the 6th Innovations in Theoretical Computer Science

Volumn , Issue , 2015, Pages 201-210

  Chekuri, Chandra ^a   Jayram, T S ^b   Vondrák, Jan ^b  

^a University of Illinois at Urbana Champaign   (United States)

^b IBM ALMADEN RESEARCH CENTER   (United States)

Author keywords

Multiplicative weight updates; Submodular function

Indexed keywords

ALGORITHMS; OPTIMIZATION;

CONTINUOUS OPTIMIZATION PROBLEMS; DISCRETIZATION PROCESS; EXTENDED ABSTRACTS; MONOTONE SUBMODULAR SET FUNCTIONS; OBJECTIVE FUNCTIONS; PACKING CONSTRAINTS; SUBMODULAR FUNCTIONS; WEIGHT UPDATE;

CONTINUOUS TIME SYSTEMS;

EID: 84961323649     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/2688073.2688086     Document Type: Conference Paper

References (40)

1. S. Arora, E. Hazan, and S. Kale. Fast algorithms for approximate semidenite programming using the multiplicative weights update method. In Proc. 46th FOCS, pages 339-348. IEEE Comp. Soc. Press, 2005.
2. S. Arora, E. Hazan, and S. Kale. The multiplicative weights update method: A meta-algorithm and applications. Theory of Computing, 8(6):121-164, 2012.
3. S. Arora and S. Kale. A combinatorial, primal-dual approach to semidenite programs. In Proc. 39th STOC, pages 227-236. ACM Press, 2007.
4. Y. Azar and I. Gamzu. Eficient submodular function maximization under linear packing constraints. In A. Czumaj, K. Mehlhorn, A. M. Pitts, and R. Wattenhofer, editors, ICALP (1), volume 7391 of Lecture Notes in Computer Science, pages 38-50. Springer, 2012.
5. A. Badanidiyuru, B. Mirzasoleiman, A. Karbasi, and A. Krause. Streaming submodular maximization: Massive data summarization on the y. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '14, pages 671-680. ACM, 2014.
6. A. Badanidiyuru and J. Vondrak. Fast algorithms for maximizing submodular functions. In SODA, pages 1497-1514, 2014.
7. N. Bansal, N. Korula, V. Nagarajan, and A. Srinivasan. Solving packing integer programs via randomized rounding with alterations. Theory of Computing, 8(1):533-565, 2012.
8. D. Bienstock. Potential Function Methods for Approximately Solving Linear Programming Problems: Theory and Practice. Springer, 2002.
9. C. Borgs, M. Brautbar, J. T. Chayes, and B. Lucier. Maximizing social inuence in nearly optimal time. In SODA, pages 946-957. SIAM, 2014.
10. G. Calinescu, C. Chekuri, M. Pal, and J. Vondrak. Maximizing a submodular set function subject to a matroid constraint (extended abstract). In M. Fischetti and D. P. Williamson, editors, IPCO, volume 4513 of Lecture Notes in Computer Science, pages 182-196. Springer, 2007.
11. G. Calinescu, C. Chekuri, M. Pal, and J. Vondrak. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput., 40(6):1740-1766, 2011.
12. C. Chekuri, J. Vondrak, and R. Zenklusen. Dependent randomized rounding via exchange properties of combinatorial structures. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 575-584. IEEE, 2010.
13. C. Chekuri, J. Vondrak, and R. Zenklusen. Submodular function maximization via the multilinear relaxation and contention resolution schemes. In L. Fortnow and S. P. Vadhan, editors, STOC, pages 783-792. ACM, 2011.
14. F. Chierichetti, R. Kumar, and A. Tomkins. Max-cover in map-reduce. In M. Rappa, P. Jones, J. Freire, and S. Chakrabarti, editors, WWW, pages 231-240. ACM, 2010.
15. F. Diedrich and K. Jansen. Faster and simpler approximation algorithms for mixed packing and covering problems. Theoretical computer science, 377(1):181-204, 2007.
16. M. Feldman, J. Naor, and R. Schwartz. A unified continuous greedy algorithm for submodular maximization. In R. Ostrovsky, editor, FOCS, pages 570-579. IEEE, 2011.
17. L. Fleischer. Approximating fractional multicommodity ow independent of the number of commodities. SIAM Journal on Discrete Mathematics, 13(4):505-520, 2000.
18. N. Garg and R. Khandekar. Fractional covering with upper bounds on the variables: Solving LPs with negative entries. In Proc. 12th Ann. Europ. Symp. on Algorithms (ESA'04), pages 371-382, 2004.
19. N. Garg and J. Konemann. Faster and simpler algorithms for multicommodity ow and other fractional packing problems. In Proc. 39th FOCS, pages 300-309. IEEE Comp. Soc. Press, 1998.
20. N. Garg and J. Konemann. Faster and simpler algorithms for multicommodity ow and other fractional packing problems. SIAM J. Comput., 37:630-652, 2007.
21. J. Gillenwater, A. Kulesza, and B. Taskar. Near-optimal map inference for determinantal point processes. In Advances in Neural Information Processing Systems, pages 2735-2743, 2012.
22. M. Grigoriadis and L. Khachiyan. Fast approximation schemes for convex programs with many blocks and coupling constraints. SIAM Journal on Optimization, 4(1):86-107, 1994.
23. M. D. Grigoriadis, L. G. Khachiyan, L. Porkolab, and J. Villavicencio. Approximate max-min resource sharing for structured concave optimization. SIAM Journal on Optimization, 11(4):1081-1091, 2001.
24. R. Iyer, S. Jegelka, and J. Bilmes. Fast semidifferential-based submodular function optimization. In Proceedings of The 30th International Conference on Machine Learning, pages 855-863, 2013.
25. K. Jansen and H. Zhang. Approximation algorithms for general packing problems and their application to the multicast congestion problem. Mathematical Programming, 114(1):183-206, 2008.
26. S. Kale. Eficient Algorithms Using The Multiplicative Weights Update Method. PhD thesis, Princeton University, 2006. Technical Report TR-804-07.
27. R. Khandekar. Lagrangian Relaxation based Algorithms for Convex Programming Problems. PhD thesis, Indian Institute of Technology, Delhi, 2004.
28. J. Konemann. Fast combinatorial algorithms for packing and covering problems. Master's thesis, Universitat des Saarlandes, 1998. Available at: http://www. math. uwaterloo. ca/ jochen/docs/diplom. ps.
29. C. Koufogiannakis and N. E. Young. Beating simplex for fractional packing and covering linear programs. In FOCS, pages 494-504. IEEE Computer Society, 2007.
30. A. Kulik, H. Shachnai, and T. Tamir. Approximations for monotone and nonmonotone submodular maximization with knapsack constraints. Mathematics of Operations Research, 38(4):729-739, 2013.
31. R. Kumar, B. Moseley, S. Vassilvitskii, and A. Vattani. Fast greedy algorithms in mapreduce and streaming. In G. E. Blelloch and B. Vocking, editors, SPAA, pages 1-10. ACM, 2013.
32. J. Kwon and P. Mertikopoulos. A continuous-time approach to online optimization. arXiv preprint arXiv:1401. 6956, 2014.
33. J. Lee, V. S. Mirrokni, V. Nagarajan, and M. Sviridenko. Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM J. Discrete Math., 23(4):2053-2078, 2010.
34. M. Luby and N. Nisan. A parallel approximation algorithm for positive linear programming. In ACM STOC, pages 448-457, 1993.
35. S. A. Plotkin, D. B. Shmoys, and fiE. Tardos. Fast approximation algorithm for fractional packing and covering problems. Math. Oper. Res., 20:257-301, 1995. Preliminary version in FOCS'91.
36. S. Sorin. Exponential weight algorithm in continuous time. Mathematical Programming, 116(1-2):513-528, 2009.
37. J. Vondrak. Optimal approximation for the submodular welfare problem in the value oracle model. In C. Dwork, editor, STOC, pages 67-74. ACM, 2008.
38. N. E. Young. Randomized rounding without solving the linear program. In Proc. 6th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA'95), pages 170-178. ACM Press, 1995.
39. N. E. Young. Sequential and parallel algorithms for mixed packing and covering. In Proceedings of 42nd IEEE Symposium on Foundations of Computer Science, pages 538-546. IEEE, 2001.
40. N. E. Young. Nearly linear-time approximation schemes for mixed packing/covering and facility-location linear programs. ArXiv e-prints, July 2014.