Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer (Z) coefficients, we show the following (Theorem 18):
Given a weighted simplicial complex and a nontrivial cycle, REF computes its homologous cycle with minimal weight.
Optimal homologous cycles, total unimodularity, and linear programming
STOC '10
Computer Science Mathematics