showing that DMSFN is NP-complete as well. The default optimization version of MSFN consequently turns out to be NP-hard. Theorem 2.2.3. $_{DMSFN}$is NP-complete, and MSFN is NP-hard. Proof. DMSFN lies in NP as we can calculate f $^{λ}$( T, S ) in polynomial time when given a pair of d - and ( d + 1)-chains t and s , respectively, of the simplicial complex K . On the other hand, given an instance of the optimal homologous chain decision problem, we can reduce it to the DMSFN by taking λ = 0 and w$_{i}$ = 1 for 1 ≤ i ≤ m . Since the optimal homologous chain problem was recently shown to be NP-complete [16, Theorem 1.4], the result follows. Remark 2.2.4. Although we showed MSFN is NP-hard in general, the case for any particular λ > 0 is not known. For λ large enough, the problem in fact becomes easy– when the ( d +1) -simplices have positive volumes and λ > ( $^{∑}$w$_{i}$ ) / min v$_{j}$ , then optimality occurs when s is the empty ( d + 1) -chain. We now consider attacking the multiscale simplicial flat norm problem using tech- niques from the area of discrete optimization. Even though the problem is NP-hard, this approach helps us to identify special cases in which we can compute the multiscale simplicial flat norm in polynomial time. 22