norm is NP-complete by reducing the optimal bounding chain problem (OBCP), which was recently shown to be NP-complete [16], to a special case of the multiscale simplicial flat norm problem. We cast the problem of finding the optimal S , and thus calculating the multiscale simplicial flat norm, as an integer linear programming (ILP) problem. Given that the original problem is NP-complete, instances of this ILP could be hard to solve. Utilizing recent work [14] on the related optimal homologous chain problem (OHCP), we provide conditions on K under which this ILP problem can in fact be solved in polynomial time. In particular, the multiscale simplicial flat norm can be computed in polynomial time when T is d -dimensional, and K is ( d + 1)-dimensional and orientable, for all d ≥ 0. A similar result holds for the case when T is d -dimensional, and K is ( d + 1)-dimensional and embedded in R d $^{+1}$, for all d ≥ 0. Our most significant contribution is the simplicial deformation theorem (Theorem 2.5.1), which states that given an arbitrary d -current in | K | (underlying space), we are assured of an approximating current in the d -skeleton of K . This result is a substantial modification and generalization of the classical deformation theorem for currents on to square grids. Our deformation theorem explicitly specifies the dependence of the bounds of approximation on the regularity and size of the simplices in the simplicial complex. Hence it is immediate from the theorem that as we refine the simplicial complex K while preserving the bounds on simplicial regularity, the flat norm distance between an arbitrary d -current in | K | and its deformation onto the d -skeleton of K vanishes. More importantly, such refinement of K does not affect the efficient computability of the multiscale simplicial flat norm by solving the associated ILP in many cases, e.g., when K is orientable or when it is full-dimensional. 15