Complexities for generalized models of self-assembly

TLDR: This paper considers whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model, and investigates the problem of verifying whether a given tile system uniquely assembles into a given shape, and shows that this problem is NP-hard.

Abstract: In this paper, we extend Rothemund and Winfree's examination of the tile complexity of tile self-assembly [6]. They provided a lower bound of Ω(log N/log log N) on the tile complexity of assembling an N × N square for almost all N. Adleman et al. [1] gave a construction which achieves this bound. We consider whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model. One of our results is a tile set of size O(√log N) which assembles an N × N square in a model which allows flexible glue strength between non-equal glues (This was independently discovered in [3]). This result is matched by a lower bound dictated by Kolmogorov complexity. For three other generalizations, we show that the Ω(log N/log log N) lower bound applies to N × N squares. At the same time, we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets. Specifically, for thin rectangles with length N and width k, we provide a tighter lower bound of Ω(N(1/k)/k) for the standard model, yet we also give a construction which achieves O(log N/log log N) complexity in a model in which the temperature of the tile system is adjusted during assembly. We also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape, and show that this problem is NP-hard.