Robust self-assembly of graphs

Stanislav Angelov, Sanjeev Khanna, Mirkó Visontai
2009 Natural Computing  
Self-assembly is a process in which small building blocks interact autonomously to form larger structures. A recently studied 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
more » ... inus 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 selfassembly 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. Rothemund and Winfree [9] proposed the Tile Assembly Model to formalize and facilitate the theoretical study of the self-assembly process. This model extends the tiling models based on Wang tiles [10] . In their work, the building blocks, namely the DNA tiles, are abstracted as oriented unit squares. Each side of a tile has a glue type and a (non-negative) strength associated to it. An assembly starts from a designated seed tile and can be augmented by a tile if the sides of the tile match the glue types of its already assembled neighbors, and the total glue strength is no less than a threshold parameter τ , referred to as the temperature of the assembly. Reif, Sahu, and Yin [11] introduced a generalization of the Tile Assembly Model, to one on general graphs, called the Accretive Graph Assembly Model. The accretive graph assembly is a sequential process where a given weighted graph is assembled one vertex at a time starting from a designated seed vertex. The weight of each positive (resp. negative) edge specifies the magnitude of attraction (resp. repulsion) between the adjacent vertices. It is feasible to add a vertex to the assembly if the total attraction minus the total repulsion of the already built neighbors is at least the temperature τ . Here, accretive suggests the monotone property of the process, i.e., once a vertex is added it cannot be removed later (cf. the Self-Destructive Graph Assembly Model [11] and the Kinetic Tile Assembly Model where tiles can fall off [12, 13] ). The Accretive Graph Assembly Model addresses some of the deficiencies of the Tile Assembly Model. For example, it models repulsion and allows the assembly of general graph structures. A central problem in this model is the Accretive Graph Assembly Problem (AGAP): Given a weighted graph, a seed vertex, and an assembly temperature τ determine if there is a sequence of feasible vertex additions that builds the graph. Among other results, Reif et al. [11] showed that AGAP is NP-complete for graphs with maximum degree 4 and for planar graphs (Planar AGAP) with maximum degree 5. Subsequently, Angelov, Khanna, and Visontai [14] improved these results by giving a dichotomy theorem which completely characterized the complexity of Planar AGAP on graphs with maximum degree 3 and only 2 possible edge weights. Specifically, it was shown that whenever the allowed edge weights and τ satisfied a simple set of inequalities the problem is NP-complete, and poly-time solvable otherwise. A drawback of the Accretive Graph Assembly Model is that even when there exists a feasible order of vertex additions to build the graph, its realization in practice may require a careful control over the order of assembly. Such control is arguably hard to implement at the molecular level, and perhaps, even in conflict with the notion of self -assembly. To alleviate this drawback, Reif et al. [11] considered a probabilistic variant of the model where at any point of time, the vertex to be build is chosen uniformly at random from the set of all vertices that can be added at that time to the partial assembly. Note that assembly still proceeds by adding one vertex at a time (cf. insufficient attachment in [12, 13] ). One of the main problems in this, so-called Stochastic Accretive Graph Assembly Model is to determine the probability of a graph system being assembled. One approach to estimating this probability is to consider the ratio of the number of
doi:10.1007/s11047-009-9149-5 fatcat:itn6qcrxcjgmlgwwg6ehxtvn5y