Abstract
This is a brief survey concerning the problem of computing optimal cycles of homology groups through linear optimization. While homology groups encode information about the presence of topological features such as holes and voids of some geometrical structure, optimal cycles tighten the representatives of the homology classes. This allows us to infer additional information concerning the location of those topological features. Moreover, by a slight modification of the original problem, we extend it to the case where we have multiple nonhomologous cycles. By considering a more general class of combinatorial structures called complexes, we recast this multiple nonhomologous cycles problem as a single cycle optimization problem in a modified complex. Finally, as a numerical example, we apply the optimal cycles problem to the 3D structure of human deoxyhemoglobin.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
H.M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T.N. Bhat, H. Weissig, I.N. Shindyalov, P.E. Bourne, The protein data bank. Nucleic Acids Res. 28(1), 235–242 (2000)
Cgal, Computational geometry algorithms library. http://www.cgal.org
C. Chen, D. Freedman, Quantifying homology classes, in Symposium on Theoretical Aspects of Computer, Science, 169–180 (2008)
C. Chen, D. Freedman, Measuring and computing natural generators for homology groups. Comput. Geom. 43(2), 169–181 (2010)
CHomP homology software. http://chomp.rutgers.edu/
T.K. Dey, A.N. Hirani, B. Krishnamoorthy, Optimal homologous cycles, total unimodularity, and linear programming. SIAM J. Comput. 40(4), 1026–1044 (2011)
H. Edelsbrunner, Weighted alpha shapes. Technical report, Department of Computer Science, University of Illinois at Urbana-Champaign, 1992
G. Fermi, M.F. Perutz, B. Shaanan, R. Fourme, The crystal structure of human deoxyhaemoglobin at 1.74 a resolution. J. Mol. Biol. 175(2), 159–174 (1984)
M. Gameiro, Y. Hiraoka, S. Izumi, M. Kramar, K. Mischaikow, V. Nanda, Topological Measurement of Protein Compressibility via Persistence Diagrams, MI Preprint Series, 6 (2012)
International Business Machines Corp, IBM ILOG CPLEX optimization studio. http://www-03.ibm.com/software/products/en/ibmilogcpleoptistud/
T. Kaczynski, K. Mischaikow, M. Mrozek, Computational Homology (Springer, New York, 2004)
J. Munkres, Elements of Algebraic Topology (The Benjamin/Cummings Publishing Company Inc, Menlo Park, 1984)
A. Schrijver, in Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics and Optimzation (Wiley, New York, 2000)
A.F. Veinott Jr, G.B. Dantzig, Integral extreme points. SIAM Rev. 10(3), 371–372 (1968)
A. Zomorodian, G. Carlsson, Computing persistent homology. Discrete Comput. Geom. 33, 249–274 (2004)
A. Zomorodian, G. Carlsson, Localized Homology. Comput. Geom. 41(3), 126–148 (2008)
Acknowledgments
The authors would like to thank Hayato Waki for valuable comments and discussions and for introducing optimization software.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Japan
About this chapter
Cite this chapter
Escolar, E.G., Hiraoka, Y. (2014). Computing Optimal Cycles of Homology Groups. In: Nishii, R., et al. A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry, vol 5. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55060-0_8
Download citation
DOI: https://doi.org/10.1007/978-4-431-55060-0_8
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-55059-4
Online ISBN: 978-4-431-55060-0
eBook Packages: EngineeringEngineering (R0)