The Complexity of Multiple Handed Self-assembly

Abstract

In this paper we study complexities for the multiple-handed tile self-assembly model, a generalization of the two-handed tile assembly model in which assembly proceeds by repeatedly combining up to h assemblies together into larger assemblies. We first show that there exist shapes that are self-assembled with provably lower tile type complexities given more hands: we construct a class of shapes $S_k$ that requires $\varOmega (\frac{k}{h})$ tile types to self-assemble with h or fewer hands, and yet is self-assembled in $\mathcal {O}(1)$ tile types with k hands. We further examine the complexity of self-assembling the classic benchmark $n\times n$ square shape, and show how this is self-assembled in $\mathcal {O}(1)$ tile types with $\mathcal {O}(n)$ hands. We next explore the complexity of established verification problems. We show the problem of determining if a given assembly is produced by an h-handed system is polynomial time solvable, whereas the problem of unique assembly verification is coNP-complete if the hand parameter h is encoded in unary, and coNEXP-complete if h is encoded in binary.

This research was supported in part by National Science Foundation Grant CCF-1817602.