Computing Optimal Cycles of Homology Groups
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.
KeywordsComputational topology Homology groups Optimal cycles
The authors would like to thank Hayato Waki for valuable comments and discussions and for introducing optimization software.
- 2.Cgal, Computational geometry algorithms library. http://www.cgal.org
- 3.C. Chen, D. Freedman, Quantifying homology classes, in Symposium on Theoretical Aspects of Computer, Science, 169–180 (2008)Google Scholar
- 5.CHomP homology software. http://chomp.rutgers.edu/
- 7.H. Edelsbrunner, Weighted alpha shapes. Technical report, Department of Computer Science, University of Illinois at Urbana-Champaign, 1992Google Scholar
- 9.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)Google Scholar
- 10.International Business Machines Corp, IBM ILOG CPLEX optimization studio. http://www-03.ibm.com/software/products/en/ibmilogcpleoptistud/
- 13.A. Schrijver, in Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics and Optimzation (Wiley, New York, 2000)Google Scholar