Computational Complexity of Determining the Assembly Index

TL;DR

This paper proves that calculating the assembly index, which measures the minimal steps to build an object from basic parts, is NP-complete and hard to approximate, highlighting its computational intractability.

Contribution

It establishes the NP-completeness and hardness of approximation for the assembly index problem, linking it to classical grammar-based compression complexity.

Findings

Decision version is NP-complete.

Optimization version is NP- and APX-hard.

Computing or approximating the assembly index is intractable.

Abstract

The assembly index of assembly theory quantifies the minimal number of composition steps required to construct an object from elementary components. The study proves that the decision version of the assembly index problem is NP-complete, through an explicit correspondence between assembly plans and straight-line grammars. This correspondence implies that the optimization version of the assembly index problem inherits NP- and APX-hardness from the classical smallest grammar problem. The study provides complete, self-contained proofs for both decision and optimization variants of the assembly index problem. These results establish that computing or approximating the assembly index is computationally intractable, placing it within the same complexity class as grammar-based compression.