The Complexity of Fixed-Height Patterned Tile Self-Assembly

TL;DR

This paper investigates the computational complexity of the PATS problem for fixed-height patterns in tile self-assembly, establishing NP-completeness results and efficient algorithms depending on pattern parameters.

Contribution

It provides the first complexity classification for fixed-height PATS patterns, including new NP-completeness proofs and polynomial-time algorithms for specific cases.

Findings

NP-completeness for height 2 or more in both variants

O(n)-time algorithms for height 1 patterns

NP-completeness persists when height and colors are fixed in the uniform variant

Abstract

We characterize the complexity of the PATS problem for patterns of fixed height and color count in variants of the model where seed glues are either chosen or fixed and identical (so-called non-uniform and uniform variants). We prove that both variants are NP-complete for patterns of height 2 or more and admit O(n)-time algorithms for patterns of height 1. We also prove that if the height and number of colors in the pattern is fixed, the non-uniform variant admits a O(n)-time algorithm while the uniform variant remains NP-complete. The NP-completeness results use a new reduction from a constrained version of a problem on finite state transducers.