3-color Bounded Patterned Self-assembly

TL;DR

This paper proves that the problem of minimal tile set synthesis for self-assembling patterns remains NP-complete even with only three colors when bounds on tile types are allowed, providing a simpler proof for the original NP-completeness result.

Contribution

It introduces multiple bound PATS (mbPATS), showing NP-completeness for patterns with just three colors, and offers a more concise proof of PATS NP-completeness.

Findings

mbPATS is NP-complete for 3-color patterns

A new, simpler proof of PATS NP-completeness

Establishes complexity bounds with minimal colors

Abstract

Patterned self-assembly tile set synthesis PATS is the problem of finding a minimal tile set which uniquely self-assembles into a given pattern. Czeizler and Popa proved the NP-completeness of PATS and Seki showed that the PATS problem is already NP-complete for patterns with 60 colors. In search for the minimal number of colors such that PATS remains NP-complete, we introduce multiple bound PATS (mbPATS) where we allow bounds for the numbers of tile types of each color. We show that mbPATS is NP-complete for patterns with just three colors and, as a byproduct of this result, we also obtain a novel proof for the NP-completeness of PATS which is more concise than the previous proofs.