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• Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the space of currents, and works by decomposing a $d$-dimensional current into $d$- and (the boundary of) $(d+1)$-dimensional pieces in an optimal way. Given an integral current, can we expect its flat norm decomposition to be integral as well? This is not known in general, except in the case of $d$-currents that are boundaries of $(d+1)$-currents in $\mathbb{R}^{d+1}$ (following results from a corresponding problem on the $L^1$ total variation ($L^1$TV) of functionals). On the other hand, for a discretized flat norm on a finite simplicial complex, the analogous statement holds even when the inputs are not boundaries. This simplicial version relies on the total unimodularity of the boundary matrix of the simplicial complex -- a result distinct from the $L^1$TV approach. We develop an analysis framework that extends the result in the simplicial setting to one for $d$-currents in $\mathbb{R}^{d+1}$, provided a suitable triangulation result holds. In $\mathbb{R}^2$, we use a triangulation result of Shewchuk (bounding both the size and location of small angles), and apply the framework to show that the discrete result implies the continuous result for $1$-currents in $\mathbb{R}^2$.
• 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.