Complexities for Generalized Models of Self-Assembly

In this paper, we study the complexity of self-assembly under models that are natural generalizations of the tile self-assembly model. In particular, we extend Rothemund and Winfree's study of the tile complexity of tile self-assembly [Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, Portland, OR, 2000, pp. 459--468]. They provided a lower bound of $\Omega(\frac{\log N}{\log\log N})$ on the tile complexity of assembling an $N\times N$ square for almost all N. Adleman et al. [Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Heraklion, Greece, 2001, pp. 740--748] 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 $O(\sqrt{\log N})$ which assembles an $N\times N$ square in a model which allows flexible glue strength between nonequal glues. This result is matched for almost all N by a lower bound dictated by Kolmogorov complexity. For three other generalizations, we show that the $\Omega(\frac{\log N}{\log\log N})$ lower bound applies to $N\times N$ 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 $\Omega(\frac{N^{1/k}}{k})$ for the standard model, yet we also give a construction which achieves $O(\frac{\log N}{\log\log N})$ 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; we show that this problem is NP-hard for three of the generalized models.