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(log N/log log N) on the tile complexity of assembling an N x 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(root log N) which assembles an N x 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 O( logN log logN) lower bound applies to N x 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(N-1/k/k) for the standard model, yet we also give a construction which achieves O( logN log logN) 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.