Optimal homologous cycles, total unimodularity, and linear programming

Proceedings of the Annual ACM Symposium on Theory of Computing

Volumn , Issue , 2010, Pages 221-230

  Dey, Tamal K ^a   Hirani, Anil N ^b   Krishnamoorthy, Bala ^c  

^a The Ohio State University   (United States)

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

^c WASHINGTON STATE UNIVERSITY   (United States)

Author keywords

computational topology; homology localization; optimal chain; simplicial complex

Indexed keywords

COMPUTATIONAL TOPOLOGY; INTEGER PROGRAMMING PROBLEMS; INTEGER SOLUTIONS; LINEAR PROGRAMS; MATRIX; NP-HARD; OPTIMIZATION APPROACH; OPTIMIZATION PROBLEMS; ORIENTABLE; POLYNOMIAL-TIME; SIMPLICIAL COMPLEX; SUBCOMPLEXES; TOTAL UNIMODULARITY; UNIMODULAR;

COMPUTATION THEORY; LINEAR PROGRAMMING; LINEARIZATION; OPTIMIZATION; POLYNOMIAL APPROXIMATION; TOPOLOGY;

INTEGER PROGRAMMING;

EID: 77954735165     PISSN: 07378017     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1806689.1806721     Document Type: Conference Paper

References (23)

1. D. Bertsimas and J. N. Tsitsiklis. Introduction to Linear Optimization. Athena Scientific, Belmont, MA., 1997.
2. E. W. Chambers, É. C. de Verdière, J. Erickson, F. Lazarus, and K. Whittlesey. Splitting (complicated) surfaces is hard. Comput. Geom. Theory Appl., 41:94-110, 2008.
3. E. W. Chambers, J. Erickson, and A. Nayyeri. Minimum cuts and shortest homologous cycles. In 25th Ann. Sympos. Comput. Geom., pages 377-385, 2009.
4. C. Chen and D. Freedman. Hardness results for homology localization. In ACM/SIAM Sympos. Discrete Algorithms, 2010. to appear.
5. C. Chen and D. Freedman. Measuring and computing natural generators for homology groups. Computational Geometry, 43(2):169-181, 2010. Special Issue on the 24th European Workshop on Computational Geometry (EuroCG'08).
6. É. C. de Verdìere and J. Erickson. Tightening non-simple paths and cycles on surfaces. In Proc. 17th Ann. ACM-SIAM Sympos. Discrete Algorithms, pages 192-201, 2006.
7. V. de Silva and R. Ghrist. Homological sensor networks. Notices of the American Mathematical Society, 54(1):10-17, 2007.
8. T. K. Dey, K. Li, J. Sun, and D. Cohen-Steiner. Computing geometry-aware handle and tunnel loops in 3d models. ACM Trans. Graph., 27:1-9, 2008.
9. T. K. Dey, J. Sun, and Y. Wang. Approximating loops in a shortest homology basis from point data. arXiv:0909.5654v2[cs.CG], page Online, 2009.
10. H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511-533, 2002.
11. J. Erickson and K. Whittlesey. Greedy optimal homotopy and homology generators. In Proc. 16th Ann. ACM-SIAM Sympos. Discrete Algorithms, pages 1038-1046, 2005.
12. O. Güler, D. den Hertog, C. Roos, T. Terlaky, and T. Tsuchiya. Degeneracy in interior point methods for linear programming: a survey. Annals of Operations Research, 46-47(1):107-138, March 1993.
13. I. Heller and C. B. Tompkins. An extension of a theorem of Dantzig's. In H. W. Kuhn and A. W. Tucker, editors, Linear Inequalities and Related Systems, Annals of Mathematics Studies, no. 38, pages 247-254. Princeton University Press, Princeton, N.J., 1956.
14. J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley Publishing Company, Menlo Park, 1984.
15. A. Schrijver. Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons Ltd., Chichester, 1986. A Wiley-Interscience Publication.
16. P. D. Seymour. Decomposition of regular matroids. J. Combin. Theory Ser. B, 28(3):305-359, 1980.
17. H. J. S. Smith. On systems of linear indeterminate equations and congruences. Philosophical Transactions of the Royal Society of London, 151:293-326, 1861.
18. E. H. Spanier. Algebraic Topology. McGraw-Hill Book Co., New York, 1966.
19. J. Sullivan. A crystalline approximation theorem for hypersurfaces. PhD thesis, Department of Mathematics, Princeton University, 1990.
20. A. Tahbaz-Salehi and A. Jadbabaie. Distributed coverage verification algorithms in sensor networks without location information. IEEE Transactions on Automatic Control, forthcoming, 55(8), August 2010.
21. E. Tardos. A strongly polynomial minimum cost circulation algorithm. Combinatorica, 5(3):247-255, September 1985.
22. E. Tardos. A strongly polynomial algorithm to solve combinatorial linear programs. Operataions Research, 34(2):250-256, March 1986.
23. A. F. Veinott, Jr. and G. B. Dantzig. Integral extreme points. SIAM Review, 10(3):371-372, 1968.