Complexities for high-temperature two-handed tile self-assembly

Abstract

Tile self-assembly is a formal model of computation capturing DNA-based nanoscale systems. Here we consider the popular two-handed tile self-assembly model or 2HAM. Each 2HAM system includes a temperature parameter, which determines the threshold of bonding strength required for two assemblies to attach. Unlike most prior study, we consider general temperatures not limited to small, constant values. We obtain two results. First, we prove that the computational complexity of determining whether a given tile system uniquely assembles a given assembly is coNP-complete, confirming a conjecture of Cannon et al. (2013). Second, we prove that larger temperature values decrease the minimum number of tile types needed to assemble some shapes. In particular, for any temperature (formula presented), we give a class of shapes of size n such that the ratio of the minimum number of tiles needed to assemble these shapes at temperature τ and any temperature less than τ is (formula presented).