Navigating central path with electrical flows: From flows to Matchings, and back (Extended Abstract)

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

Volumn , Issue , 2013, Pages 253-262

  Ma̧dry, Aleksander ^a  

^a EPFL   (Switzerland)

Author keywords

Bipartite matchings; Central path; Electrical flows; Interior point methods; Laplacian linear systems; Maximum flow problem; Minimum s t cut problem

Indexed keywords

BIPARTITE MATCHINGS; CENTRAL PATH; ELECTRICAL FLOWS; INTERIOR-POINT METHOD; LAPLACIAN LINEAR SYSTEMS; MAXIMUM FLOW PROBLEMS; MINIMUM S-T CUT PROBLEM;

ALGORITHMS; COMPUTER SCIENCE; CONVEX OPTIMIZATION; LINEAR SYSTEMS;

GRAPH THEORY;

EID: 84893495424     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/FOCS.2013.35     Document Type: Conference Paper

References (54)

1. A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, 1st ed. Boston, MA, USA: Addison-Wesley Longman Publishing Co., 1974.
2. R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network flows: Theory, algorithms, and applications. Prentice-Hall, 1993.
3. R. K. Ahuja, T. L. Magnanti, J. B. Orlin, and M. R. Reddy, Applications of Network Optimization, ser. Handbooks in Operations Research and Management Science. North- Holland, 1995, vol. 7.
4. H. Alt, N. Blum, K. Mehlhorn, and M. Paul, "Computing a maximum cardinality matching in a bipartite graph in time O(n1.5 √m/log n), " Inf. Process. Lett., vol. 37, no. 4, pp. 237-240, 1991.
5. S. Arora, E. Hazan, and S. Kale, "The multiplicative weights update method: A meta-algorithm and applications, " Theory of Computing, vol. 8, no. 1, pp. 121-164, 2012.
6. B. Bollobas, Modern Graph Theory. Springer, 1998.
7. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
8. J. R. Bunch and J. E. Hopcroft, "Triangular factorization and inversion by fast matrix multiplication, " Mathematics of Computation, vol. 28(125), pp. 231-236, 1974.
9. P. Christiano, J. Kelner, A. Madry, D. Spielman, and S.- H. Teng, "Electrical flows, Laplacian systems, and faster approximation of maximum flow in undirected graphs, " in STOC'11: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, 2011, pp. 273-281.
10. D. Coppersmith and S. Winograd, "Matrix multiplication via arithmetic progressions, " Journal of Symbolic Computation, vol. 9, pp. 251-280, 1990.
11. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 3rd ed. The MIT Press, 2009.
12. S. I. Daitch and D. A. Spielman, "Faster approximate lossy generalized flow via interior point algorithms, " in STOC'08: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008, pp. 451-460.
13. J. Edmonds, "Paths, trees, and flowers, " Canadian Journal of Mathematics, vol. 17, pp. 449-467, 1965.
14. J. Egerváry, "Matrixok kombinatorius tulajdonsá gairól, " Matematikaiés Fizikai Lapok, vol. 38, pp. 16-28, 1931.
15. P. Elias, A. Feinstein, and C. E. Shannon, "A note on the maximum flow through a network, " IRE Transactions on Information Theory, vol. 2, 1956.
16. S. Even and R. E. Tarjan, "Network flow and testing graph connectivity, " SIAM Journal on Computing, vol. 4, no. 4, pp. 507-518, 1975.
17. T. Feder and R. Motwani, "Clique partitions, graph compression and speeding-up algorithms, " Journal of Computer and System Sciences, vol. 51(2), pp. 261-272, 1995.
18. L. R. Ford and D. R. Fulkerson, "Maximal flow through a network, " Canadian Journal of Mathematics, vol. 8, pp. 399- 404, 1956.
19. H. N. Gabow and R. E. Tarjan, "Faster scaling algorithms for general graph matching problems, " Journal of the ACM, vol. 38, no. 4, pp. 815-853, 1991.
20. A. Goel, M. Kapralov, and S. Khanna, "Perfect matchings in O(n log n) time in regular bipartite graphs, " in STOC'10: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing, 2010, pp. 39-46.
21. A. V. Goldberg and A. V. Karzanov, "Maximum skewsymmetric flows and matchings, " Mathematical Programming, vol. 100, no. 3, pp. 537-568, 2004.
22. A. V. Goldberg and S. Rao, "Beyond the flow decomposition barrier, " Journal of the ACM, vol. 45, no. 5, pp. 783-797, 1998.
23. N. J. A. Harvey, "Algebraic algorithms for matching and matroid problems, " SIAM Journal on Computing, vol. 39, no. 2, pp. 679-702, 2009.
24. A. J. Hoffman, "Some recent applications of the theory of linear inequalities to extremal combinatorial analysis, " in Proceedings of Symposia in Applied Mathematics, vol. 10, 1960, pp. 113-127.
25. J. Hopcroft and R. Karp, "An n5/2 algorithm for maximum matchings in bipartite graphs, " SIAM Journal on Computing, vol. 2, no. 4, pp. 225-231, 1973.
26. J. A. Kelner, Y. T. Lee, L. Orecchia, and A. Sidford, "An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations, " 2013. [Online]. Available: http://arxiv.org/abs/1304.2338.
27. J. A. Kelner, L. Orecchia, A. Sidford, and Z. A. Zhu, "A simple, combinatorial algorithm for solving SDD systems in nearly-linear time, " in STOC'13: Proceedings of the 45th Annual ACM Symposium on the Theory of Computing, 2013, pp. 911-920.
28. S. Khanna, T. C. Kwok, and L. C. Lau, 2013, personal communication.
29. V. King, S. Rao, and R. Tarjan, "A faster deterministic maximum flow algorithm, " Journal of Algorithms, vol. 17, no. 3, pp. 447-474, 1994.
30. D. König, "Graphokés matrixok, " Matematikaiés Fizikai Lapok, vol. 38, pp. 116-119, 1931.
31. I. Koutis, G. L. Miller, and R. Peng, "Approaching optimality for solving SDD systems, " in FOCS'10: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 235-244.
32. I. Koutis, G. L. Miller, and R. Peng, "A nearly mlog n-time solver for SDD linear systems, " in FOCS'11: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, 2011, pp. 590- 598.
33. L. C. Lau, 2013, personal communication.
34. Y. T. Lee, S. Rao, and N. Srivastava, "A new approach to computing maximum flows using electrical flows, " in STOC'13: Proceedings of the 45th Annual ACM Symposium on the Theory of Computing, 2013, pp. 755-764.
35. L. Lovász, "On determinants, matchings and random algorithms, " Fundamentals of Computation Theory, vol., 565-574, 1979.
36. L. Lovász and D. M. Plummer, Matching Theory. Elsevier Science, 1986.
37. A. Madry, "Fast approximation algorithms for cut-based problems in undirected graphs, " in FOCS'10: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 245-254.
38. A. Madry, "From graphs to matrices, and back: New techniques for graph algorithms, " Ph.D. dissertation, Massachusetts Institute of Technology, 2011.
39. A. Madry, "Navigating central path with electrical flows: from flows to matchings, and back, " 2013, available at http://arxiv.org/abs/1307. 2205.
40. S. Micali and V. V. Vazirani, "An O( √V E) algoithm for finding maximum matching in general graphs, " in FOCS'80: Proceedings of the 21st Annual IEEE Symposium on Foundations of Computer Science, 1980, pp. 17-27.
41. M. Mucha, "Finding maximum matchings via Gaussian elimination, " Ph.D. dissertation, University of Warsaw, 2005.
42. M. Mucha and P. Sankowski, "Maximum matchings via Gaussian elimination, " in FOCS'04: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004, pp. 248-255.
43. J. B. Orlin, "Max flows in o(nm) time, or better, " in STOC'13: Proceedings of the 45th Annual ACM Symposium on the Theory of Computing, 2013, pp. 765-774.
44. M. O. Rabin and V. V. Vazirani, "Maximum matchings in general graphs through randomization, " J. Algorithms, vol. 10, no. 4, pp. 557-567, Dec. 1989.
45. A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003.
46. J. Sherman, "Breaking the multicommodity flow barrier for O( √ log n)-approximations to sparsest cuts, " in FOCS'09: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, 2009, pp. 363-372.
47. J. Sherman, "Nearly maximum flows in nearly linear time, " in FOCS'13: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, 2013.
48. D. A. Spielman and S.-H. Teng, "Solving sparse, symmetric, diagonally-dominant linear systems in time O(m1.31), " in FOCS'03: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003.
49. D. A. Spielman and S.-H. Teng, "Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems, " in STOC'04: Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, 2004, pp. 81-90.
50. W. T. Tutte, "The factorization of linear graphs, " Journal of the London Mathematical Society, vol. 22, pp. 107-111, 1947.
51. V. Vassilevska Williams, "Multiplying matrices faster than Coppersmith-Winograd, " in STOC'12: Proceedings of the 44th Annual ACM Symposium on the Theory of Computing, 2012.
52. V. V. Vazirani, "A theory of alternating paths and blossoms for proving correctness of the O( √V E) general graph matching algorithms, " Combinatorica, vol. 14 (1), pp. 71- 109, 1994.
53. S. J. Wright, Primal-Dual Interior-Point Methods. Society for Industrial and Applied Mathematics, 1997.
54. Y. Ye, Interior Point Algorithms: Theory and Analysis. John Wiley & Sons, 1997.