On the complexity of graph self-assembly in accretive systems

We study the complexity of the Accretive Graph Assembly Problem (\({\tt{AGAP}}\)). An instance of \({\tt{AGAP}}\) consists of an edge-weighted graph G, a seed vertex in G, and a temperature τ. The goal is to determine if the graph G can be assembled by a sequence of vertex additions starting from the seed vertex. The edge weights model the forces of attraction and repulsion, and determine which vertices can be added to a partially assembled graph at the given temperature. A vertex can be added when the total weight to its already built neighbors in the graph is at least τ. The assembly process is sequential meaning that only one vertex can be added at a time. Our first result is that \({\tt{AGAP}}\) is NP-complete even on planar graphs with maximum degree 3 when edges have only two different types of weights. This resolves the complexity of \({\tt{AGAP}}\) in the sense that the problem is poly-time solvable when either the maximum degree is at most 2 or the number of distinct edge weights is one, and is NP-complete otherwise. Our second result is a dichotomy theorem that completely characterizes the complexity of \({\tt{AGAP}}\) on graphs with maximum degree 3 and two distinct weights: w _p and w _n . We give a simple system of linear constraints on w _p , w _n , and τ that determines whether the problem is NP-complete or is poly-time solvable. In the process of establishing this dichotomy, we give a poly-time algorithm to solve a non-trivial class of \({\tt{AGAP}}.\) Finally, we consider the optimization version of \({\tt{AGAP}}\) where the goal is to assemble a largest-possible induced subgraph of the given input graph. We show that even on graphs that can be assembled and have maximum degree 3, it is NP-hard to assemble a (1/n ^1-ε)-fraction of the input graph for any \(\varepsilon > 0;\) here n denotes the number of vertices in G.