Solving NP-complete problems in the tile assembly model

Formalized study of self-assembly has led to the definition of the tile assembly model, a highly distributed parallel model of computation that may be implemented using molecules or a large computer network such as the Internet. Previously, I defined deterministic and nondeterministic computation in the tile assembly model and showed how to add, multiply and factor. Here, I extend the notion of computation to include deciding subsets of the natural numbers, and present a system that decides SubsetSum, a well-known NP-complete problem. The computation is nondeterministic and each parallel assembly executes in time linear in the input. The system requires only a constant number of different tile types: 49. I describe mechanisms for finding the successful solutions among the many parallel assemblies and explore bounds on the probability of such a nondeterministic system succeeding and prove that probability can be made arbitrarily close to one. c © 2007 Elsevier B.V. All rights reserved.