Optimal Cycles for Persistent Homology Via Linear Programming

In this work, we discuss the problem of finding optimal cycles for homology groups of simplicial complexes and for persistent homology of filtrations. We review the linear programming formulation of the optimal homologous cycle problem and its extension to allow for multiple cycles. By inserting these linear programming problems into the persistent homology algorithm, we are able to compute an optimal cycle, that has been optimized at birth, for every persistent interval in the persistent diagram.