Optimal Homologous Cycles, Total Unimodularity, and Linear Programming

Abstract: 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 coefficients, we show the following : For a finite simplicial complex $K$ of dimension greater than $p$, the boundary matrix $[\partial_{p+1}]$ is totally unimodular if and only if $H_p(L, L_0)$ is torsion-free, for all pure subcomplexes $L_0, L$ in $K$ of dimensions $p$ and $p+1$ respectively, where $L_0$ is a subset of $L$. Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain integer solution. Thus the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under $\mathbb{Z}_2$ coefficients which, being a field, is in general easier to deal with. One consequence of our result, among others, is that one can compute in polynomial time an optimal 2-cycle in a given homology class for any finite simplicial complex embedded in $\mathbb{R}^3$. Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles.