T − ∂S . To study the complexity of computing the multiscale simplicial flat norm, we consider a decision version of the problem, termed decision -MSFN or DMSFN . The function f $^{λ}$( T, S ) used here is defined in Equation 2.6, with the modification that w$_{i}$ and v$_{j}$ are assumed to be rational for purposes of analyses of complexity. Definition 2.2.2 ( DMSFN ) . Given a p -dimensional finite simplicial complex K with p ≥ d + 1 , a set T defined as a d -subcomplex of K , a scale λ ∈ [0 , ∞ ) , and a rational number f$_{0}$ ≥ 0 , does there exist a ( d + 1) -dimensional subcomplex S of K such that f $^{λ}$( T, S ) ≤ f$_{0}$ ? The related optimal homologous chain problem (OHCP) was recently shown to be NP-complete [16, Theorem 1.4]. We reduce OHCP to a special case of DMSFN , thus 21 2.2.1 Complexity of multiscale simplicial flat norm Figure 2.2: The multiscale simplicial flat norm illustrated for two different scales λ = 1 and λ ≪ 1. See text for explanation. S ∂S T − ∂S for λ ≪ 1 T − ∂S T Ω