Computational complexity of the Assembly Index Problem

Ascertain the computational complexity of the Assembly Index Problem: given a directed hypergraph representing an assembly space and a target vertex x, compute the assembly index ass(x) defined as the minimum number of hyperedges in an assembly pathway for x; determine whether this problem is NP-hard, NP-complete, or solvable in polynomial time.

Background

The paper formalizes the assembly index as the minimum number of hyperedges in an assembly pathway for a target vertex. It notes recent results showing that the related Assembly Steps Problem (finding an optimal assembly pathway under a different objective) is NP-complete.

Despite these advances, the authors explicitly state that the computational complexity of the Assembly Index Problem itself remains open, highlighting a key unresolved question in the algorithmic analysis of assembly theory.