model of self-assembly is the Accretive Graph Assembly Model whereby an edge-weighted graph is assembled one vertex at a time
starting from a designated seed vertex. The weight of an edge specifies the magnitude of attraction (positive weight) or repulsion
(negative weight) between adjacent vertices. It is feasible to add a vertex to the assembly if the total attraction minus repulsion of the already built neighbors exceeds a certain threshold,
called the assembly temperature. This model naturally generalizes the extensively studied Tile Assembly Model. A natural question
in graph self-assembly is to determine whether or not there exists a sequence of feasible vertex additions to realize the
entire graph. However, even when it is feasible to realize the assembly, not much can be inferred about its likelihood of
realization in practice due to the uncontrolled nature of the self-assembly process. Motivated by this, we introduce the robust self-assembly problem where the goal is to determine if every possible sequence of feasible vertex additions leads to the
completion of the assembly. We show that the robust self-assembly problem is co-NP-complete even on planar graphs with two
distinct edge weights. We then examine the tractability of the robust self-assembly problem on a natural subclass of planar
graphs, namely grid graphs. We identify structural conditions that determine whether or not a grid graph can be robustly self-assembled,
and give poly-time algorithms to determine this for several interesting cases of the problem. Finally, we also show that the
problem of counting the number of feasible orderings that lead to the completion of an assembly is #P-complete.
- Content Type Journal Article
- DOI 10.1007/s11047-009-9149-5
- Authors
- Stanislav Angelov, Google, Inc. New York NY 10011 USA
- Sanjeev Khanna, University of Pennsylvania Department of Computer and Information Science, School of Engineering and Applied Sciences Philadelphia PA 19104 USA
- Mirkó Visontai, University of Pennsylvania Department of Mathematics, School of Arts and Sciences Philadelphia PA 19104 USA
- Journal Natural Computing
- Online ISSN 1572-9796
- Print ISSN 1567-7818
- Journal Volume Volume 9
- Journal Issue Volume 9, Number 1 / March, 2010