by Yuriy Brun

Abstract:

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 $\mathitSubsetSum$, 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 1.

Citation:

Yuriy Brun, Solving NP-complete problems in the tile assembly model, Theoretical Computer Science, vol. 395, no. 1, April 2008, pp. 31–46.

Related:

Extended and revised version of "Constant-size tileset for solving
an NP-complete problem in nondeterministic linear time" in DNA Computing 2008.
A previous version appeared as University of Southern California, Center for
Software Engineering technical report USC-CSSE-2007-703.

Bibtex:

@article{Brun08np-c, author = {Yuriy Brun}, title = {\href{http://people.cs.umass.edu/brun/pubs/pubs/Brun08np-c.pdf}{Solving {NP}-complete problems in the tile assembly model}}, journal = {Theoretical Computer Science}, venue = {TCS}, volume = {395}, number = {1}, pages = {31--46}, month = {April}, date = {17}, year = {2008}, issn = {0304-3975}, doi = {10.1016/j.tcs.2007.07.052}, publisher = {Elsevier}, address = {Essex, {UK}}, note = {Extended and revised version of~\ref{Brun08dna-lncs}. A previous version appeared as University of Southern California, Center for Software Engineering technical report USC-CSSE-2007-703. \href{https://doi.org/10.1016/j.tcs.2007.07.052}{DOI: 10.1016/j.tcs.2007.07.052}}, previous = {Extended and revised version of "Constant-size tileset for solving an NP-complete problem in nondeterministic linear time" in DNA Computing 2008. A previous version appeared as University of Southern California, Center for Software Engineering technical report USC-CSSE-2007-703.}, abstract = {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 $\mathit{SubsetSum}$, 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 1.}, }