  Computing an optimal cycle in a given homology class, also referred to as the
homology localization problem, is known to be an NP-hard problem in general.
Furthermore, there is currently no known optimality criterion that localizes
classes geometrically and admits a stability property under the setting of
persistent homology. We present a geometric optimization of the cycles that is
computable in polynomial time and is stable in an approximate sense. Tailoring
our search criterion to different settings, we obtain various optimization
problems like optimal homologous cycle, minimum homology basis, and minimum
persistent homology basis. In practice, the (trivial) exact algorithm is
computationally expensive despite having a worst case polynomial runtime.
Therefore, we design approximation algorithms for the above problems and study
their performance experimentally. These algorithms have reasonable runtimes for
moderate sized datasets and the cycles computed by these algorithms are
consistently of high quality as demonstrated via experiments on multiple
datasets.
