Computing Optimal Cycles of Homology Groups

Part of the Mathematics for Industry book series (MFI, volume 5)


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.


Computational topology Homology groups Optimal cycles 



The authors would like to thank Hayato Waki for valuable comments and discussions and for introducing optimization software.


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Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Graduate School of Mathematics Kyushu UniversityNishi-kuJapan
  2. 2.Institute of Mathematics for IndustryKyushu UniversityNishi-kuJapan

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