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Complexity of Graph Self-assembly in Accretive Systems and Self-destructible Systems

  • John H. Reif
  • Sudheer Sahu
  • Peng Yin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3892)

Abstract

Self-assembly is a process in which small objects autonomously associate with each other to form larger complexes. It is ubiquitous in biological constructions at the cellular and molecular scale and has also been identified by nanoscientists as a fundamental method for building nano-scale structures. Recent years see convergent interest and efforts in studying self-assembly from mathematicians, computer scientists, physicists, chemists, and biologists. However most complexity theoretic studies of self-assembly utilize mathematical models with two limitations: 1) only attraction, while no repulsion, is studied; 2) only assembled structures of two dimensional square grids are studied. In this paper, we study the complexity of the assemblies resulting from the cooperative effect of repulsion and attraction in a more general setting of graphs. This allows for the study of a more general class of self-assembled structures than the previous tiling model. We define two novel assembly models, namely the accretive graph assembly model and the self-destructible graph assembly model, and identify one fundamental problem in them: the sequential construction of a given graph, referred to as Accretive Graph Assembly Problem (\(\textsc{AGAP}\)) and Self-Destructible Graph Assembly Problem (\(\textsc{DGAP}\)), respectively. Our main results are: (i) \(\textsc{AGAP}\)  is NP-complete even if the maximum degree of the graph is restricted to 4 or the graph is restricted to be planar with maximum degree 5; (ii) counting the number of sequential assembly orderings that result in a target graph (\(\textsc{\#AGAP}\)) is #P-complete; and (iii) \(\textsc{DGAP}\)  is PSPACE-complete even if the maximum degree of the graph is restricted to 6 (this is the first PSPACE-complete result in self-assembly). We also extend the accretive graph assembly model to a stochastic model, and prove that determining the probability of a given assembly in this model is #P-complete.

Keywords

Black Vertex Negative Edge Connector Vertex Target Graph Graph Assembly 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • John H. Reif
    • 1
  • Sudheer Sahu
    • 1
  • Peng Yin
    • 1
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA

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