content_image/313631.jpg
T − ∂S . 

<img src='content_image/313631.jpg'>

## 2.2.1 Complexity of multiscale simplicial flat norm 

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 

