Minimum cuts and shortest homologous cycles

Proceedings of the Annual Symposium on Computational Geometry

Volumn , Issue , 2009, Pages 377-385

  Chambers, Erin W ^a   Erickson, Jeff ^b   Nayyeri, Amir ^b  

^a SAINT LOUIS UNIVERSITY   (United States)

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

Author keywords

Computational topology; Topological graph theory

Indexed keywords

COMPUTATIONAL TOPOLOGY; EMBEDDED GRAPHS; MINIMUM CUT; NEAR-LINEAR TIME; NP-HARD; ORIENTABLE SURFACES; PLANAR GRAPH; SINGLE INPUT; SPARSE GRAPHS; SUBGRAPHS; TIME ALGORITHMS; TIME BOUND; TOPOLOGICAL GRAPHS; UNDIRECTED GRAPH;

ALGORITHMS; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL GEOMETRY; COSTS;

GRAPH THEORY;

EID: 70849089497     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1542362.1542426     Document Type: Conference Paper

References (74)

1. R. K. Ahuja, T. L. Magnanti, and J. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993.
2. G. Borradaile. Exploiting Planarity for Network Flow and Connectivity Problems. Ph.D. thesis, Brown University, May 2008. (http://www.cs.brown.edu/ research/pubs/theses/phd/2008/glencora.pdf).
3. G. Borradaile, E. D. Demaine, and S. Tazari. Polynomialtime approximation schemes for subset-connectivity problems in bounded-genus graphs. Proc. 26th Int. Symp. Theoretical Aspects Comput. Sci., 171-182, 2009. Dagstuhl Seminar Proceedings. (http://drops.dagstuhl.de/opus/volltexte/2009/1835/).
4. G. Borradaile, C. Kenyon-Mathieu, and P. N. Klein. A polynomial-time approximation scheme for Steiner tree in planar graphs. Proc. 18th Ann. ACM-SIAM Symp. Discrete Algorithms, 1285-1294, 2007.
5. G. Borradaile, C. Kenyon-Mathieu, and P. N. Klein. Steiner tree in planar graphs: An O(n log n) approximation scheme with singly-exponential dependence on epsilon. Proc. 10th Ann. Workshop on Algorithms and Data Structures, 275-286, 2007.
6. G. Borradaile and P. Klein. An O(n log n)-time algorithm for maximum st-flow in a directed planar graph. Proc. 17th Ann. ACM-SIAM Symp. Discrete Algorithms, 524-533, 2006.
7. G. Borradaile and P. Klein. An O(n log n) algorithm for maximum st-flow in a directed planar graph. J. ACM, to appear, 2009. (http://www.math.uwaterloo. ca/~glencora/downloads/maxflow-full.pdf).
8. S. Cabello. Many distances in planar graphs. Proc. 17th Ann. ACM-SIAM Symp. Discrete Algorithms, 1213-1220, 2006.
9. S. Cabello and E. W. Chambers. Multiple source shortest paths in a genus g graph. Proc. 18th Ann. ACM-SIAM Symp. Discrete Algorithms, 89-97, 2007.
10. S. Cabello, M. DeVos, J. Erickson, and B. Mohar. Finding one tight cycle. Proc. 19th Ann. ACM-SIAM Symp. Discrete Algorithms, 527-531, 2008.
11. S. Cabello and B. Mohar. Finding shortest non-separating and non-contractible cycles for topologically embedded graphs. Discrete Comput. Geom. 37:213-235, 2007.
12. E. W. Chambers, É. Colin de Verdière, J. Erickson, F. Lazarus, and K. Whittlesey. Splitting (complicated) surfaces is hard. Proc. 22nd Ann. ACM Symp. Comput. Geom., 421-429, 2006.
13. E. W. Chambers, É. Colin de Verdière, J. Erickson, F. Lazarus, and K. Whittlesey. Splitting (complicated) surfaces is hard. Comput. Geom. Theory Appl. 41(1-2):94-110, 2008.
14. E. W. Chambers, J. Erickson, and A. Nayyeri. Homology flows, cohomology cuts. Proc. 41st Ann. ACM Symp. Theory Comput., 2009.
15. C. Chen and D. Freedman. Quantifying homology classes II: Localization and stability. Preprint, 2007. (http://arxiv.org/abs/0709.2512v2).
16. C. Chen and D. Freedman. Quantifying homology classes. Proc. 25th Ann. Symp. Theoretical Aspects Comp. Sci., 169-180, 2008. Dagstuhl Seminar Proceedings. (http://drops.dagstuhl.de/opus/volltexte/2008/1343/).
17. É. Colin de Verdière. Raccourcissement de courbes et décomposition de surfaces [Shortening of Curves and Decomposition of Surfaces]. Ph.D. thesis, University of Paris 7, Dec. 2003. (http://www.di.ens. fr/~colin/textes/these.html).
18. É. Colin de Verdière and J. Erickson. Tightening non-simple paths and cycles on surfaces. Proc. 17th Ann. ACM-SIAM Symp. Discrete Algorithms, 192-201, 2006.
19. É. Colin de Verdière and F. Lazarus. Optimal pants decompositions and shortest homotopic cycles on an orientable surface. Proc. 11th Sympos. Graph Drawing, 478-490, 2003. Lecture Notes Comput. Sci. 2912.
20. É. Colin de Verdière and F. Lazarus. Optimal system of loops on an orientable surface. Discrete Comput. Geom. 33(3):507-534, 2005.
21. É. Colin de Verdière and F. Lazarus. Optimal pants decompositions and shortest homotopic cycles on an orientable surface. J. ACM 54(4), 2007.
22. E. D. Demaine, M. Hajiaghayi, and B. Mohar. Approximation algorithms via contraction decomposition. Proc. 18th Ann. ACM-SIAM Symp. Discrete Algorithms ACM-SIAM symposium on Discrete algorithms, 278-287, 2007.
23. T. K. Dey, K. Li, and J. Sun. On computing handle and tunnel loops. IEEE Proc. Int. Conf. Cyberworlds, 357-366,2007.
24. T. K. Dey, K. Li, J. Sun, and D. Cohen-Steiner. Computing geometry-aware handle and tunnel loops in 3D models. ACM Trans. Graphics 27(3):1-9, 2008. Proc. SIGGRAPH 2008.
25. D. Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms and Applications 3(3):1-27, 1999.
26. D. Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica 27:275-291, 2000.
27. J. Erickson and S. Har-Peled. Optimally cutting a surface into a disk. Discrete Comput. Geom. 31(1):37-59, 2004.
28. J. Erickson and K. Whittlesey. Greedy optimal homotopy and homology generators. Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithms, 1038-1046, 2005.
29. J. Erickson and P. Worah. Computing the shortest essential cycle. Preprint, November 2008. (http://www.cs.uiuc.edu/~jeffe/pubs/essential.html).
30. J. Fakcharoenphol and S. Rao. Planar graphs, negative weight edges, shortest paths, and near linear time. J. Comput. Syst. Sci. 72(5):868-889, 2006.
31. L. R. Ford and D. R. Fulkerson. Maximal flow through a network. Canadian J. Math. 8(399-404), 1956.
32. G. N. Frederickson. Fast algorithms for shortest paths in planar graphs with applications. SIAM J. Comput. 16(6):1004-1004, 1987.
33. M. L. Furst, J. L. Gross, and L. A. McGeoch. Finding a maximum-genus graph imbedding. J. Assoc. Comput. Mach. 35(3):523-534, 1988.
34. A. V. Goldberg and S. Rao. Beyond the flow decomposition barrier. J. ACM 45(5):783-797, 1998.
35. A. V. Goldberg and R. E. Tarjan. A new approach to the maximum-flow problem. J. Assoc. Comput. Mach. 35(4):921-940, 1988.
36. M. Grohe. Isomorphism testing for embeddable graphs through definability. Proc. 32nd ACM Symp. Theory Comput., 63-72, 2000.
37. J. L. Gross and T. W. Tucker. Topological graph theory. Dover Publications, 2001.
38. I. Guskov and Z. Wood. Topological noise removal. Proc. Graphics Interface, 19-26, 2001.
39. T. E. Harris and F. S. Ross. Fundamentals of a method for evaluating rail net capacities. Tech. rep., The RAND Corporation, Santa Monica, California, October 24 1955. Cited in [66].
40. R. Hassin and D. B. Johnson. An O(n log2 n) algorithm for maximum flow in undirected planar networks. SIAM J. Comput. 14(3):612-624, 1985.
41. A. Hatcher. Algebraic Topology. Cambridge University Press, 2001. (http://www.math.cornell.edu/~hatcher/).
42. M. R. Henzinger, P. Klein, S. Rao, and S. Subramanian. Faster shortest-path algorithms for planar graphs. J. Comput. Syst. Sci. 55(1):3-23, 1997.
43. J. M. Hochstein and K. Weihe. Maximum s-t-flow with k crossings in O(k3n log n) time. Proc. 18th Ann. ACM-SIAM Symp. Discrete Algorithms, 843-847, 2007.
44. J. E. Hopcroft and J. K. Wong. Linear time algorithm for isomorphism of planar graphs (preliminary report). Proc. 6th ACM Symp. Theory Comput., 172-184, 1974.
45. H. Imai and K. Iwano. Efficient sequential and parallelv algorithms for planar minimum cost flow. Proc. SIGAL Int. Symp. Algorithms, 21-30, 1990. Lecture Notes Comput. Sci. 450, Springer-Verlag.
46. P. Indyk and A. Sidiropoulos. Probabilistic embeddings of bounded genus graphs into planar graphs. Proc. 23rd Ann. ACM Symp. Comput. Geom., 204-209, 2007.
47. A. Itai and Y. Shiloach. Maximum flow in planar networks. SIAM J. Comput. 8:135-150, 1979.
48. L. Janiga and V. Koubek. Minimum cut in directed planar networks. Kybernetika 28(1):37-49, 1992.
49. K. Kawarabayashi and B. Reed. Computing crossing number in linear time. Proc. 39th Ann ACM Symp. Theory Comput., 382-390, 2007.
50. P. Klein. Multiple-source shortest paths in planar graphs. Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithms, 146-155, 2005.
51. 2 n)-time algorithm. Proc. 20th Ann. ACMSIAM Symp. Discrete Algorithms, 236-245, 2009.
52. M. Kutz. Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time. Proc. 22nd Ann. ACM Symp. Comput. Geom., 430-438, 2006.
53. R. J. Lipton, D. J. Rose, and R. E. Tarjan. Generalized nested dissection. SIAM J. Numer. Anal. 16:346-358, 1979.
54. M. Mareš. Two linear time algorithms for MST on minor closed graph classes. Archivum Mathematicum 40(3):315-320, 2004.
55. S. T. McCormick, M. R. Rao, and G. Rinaldi. Easy and difficult objective functions for max cut. Math. Program., Ser. B 94:459-466, 2003.
56. G. L. Miller. Isomorphism testing for graphs of bounded genus. Proc. 12th ACM Symp. Theory Comput., 225-235, 1980.
57. G. L. Miller and J. Naor. Flow in planar graphs with multiple sources and sinks. SIAM J. Comput. 24(5):1002-10017, 1995.
58. B. Mohar and C. Thomassen. Graphs on Surfaces. Johns Hopkins University Press, 2001.
59. K. Mulmuley, U. Vazirani, and V. Vazirani. Matching is as easy as matrix inversion. Combinatorica 7:105-113, 1987.
60. V. Y. Pan and J. H. Reif. Fast and efficient parallel solution of sparse linear systems. SIAM J. Comput. 22(6):1227-1250, 1993.
61. D. Pe'er. On minimum spanning trees. Master's thesis, Hebrew University, 1998. (http://www.math.ias.edu/~avi/STUDENTS/dpthesis.pdf).
62. 2 n) time. SIAM J. Comput. 12:71-81, 1983.
63. G. Ringel. Map Color Theorem. Springer-Verlag, 1974.
64. G. Ringel and J. W. T. Youngs. Solution of the Heawood map-coloring problem. Proc. Nat. Acad. Sci. USA 60:438-445, 1968.
65. A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics 24. Springer-Verlag, 2003.
66. A. Schrijver. On the history of combinatorial optimization (till 1960). Handbook of Discrete Optimization, 1-68, 2005. Elsevier.
67. D. D. Sleator and R. E. Tarjan. A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3):362-391, 1983.
68. J. Stillwell. Classical Topology and Combinatorial Group Theory, 2nd edition. Graduate Texts in Mathematics 72. Springer-Verlag, 1993.
69. S. Tazari and M. Müller-Hannemann. Shortest paths in linear time on minor-closed graph classes, with an application to Steiner tree approximation. Discrete Appl. Math. 157:673-684, 2009.
70. C. Thomassen. Embeddings of graphs with no short noncontractible cycles. J. Comb. Theory Ser. B 48(2):155-177,1990.
71. P. M. Vaidya. Speeding-up linear programming using fast matrix multiplication. Proc. 30th IEEE Symp. Found. Comput. Sci., 332-337, 1989.
72. K. Weihe. Maximum (s, t)-flows in planar networks in O(|V|log|V|)-time. J. Comput. Syst. Sci. 55(3):454-476, 1997.
73. A. Zomorodian. Topology and Computing. Cambridge University Press, 2005.
74. A. Zomorodian and G. Carlsson. Localized homology. Proc. IEEE Int Conf. Shape Modeling Appl. (SMI), 189-198, 2007.