On the complexity of graph self-assembly in accretive systems

Abstract

We study the complexity of the Accretive Graph Assembly Problem (AGAP). An instance of AGAP consists of an edge-weighted graph G, a seed vertex in G, and a temperature τ. The goal is to determine if there is a sequence of vertex additions which constructs G starting from the seed. 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. Our first result is that AGAP is NP-complete even on degree 3 planar graphs when edges have only two different types of weights. This resolves the complexity of AGAP in the sense that the problem is polytime solvable when either the degree is bounded by 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 AGAP on degree 3 bounded graphs with two distinct weights: w p, wn. We give a simple system of linear constraints on wp, wn, and τ that determines whether the problem is NP-complete or is polytime solvable. In the process of establishing this dichotomy, we give the first polytime algorithm to solve a non-trivial class of AGAP Finally, we consider the optimization version of AGAP where the goal is to realize a largest-possible subgraph of the given input graph. We show that even on constructible graphs of degree at most 3, it is NP-hard to realize a (1/n1-ε)-fraction of the input graph for any ε> 0; here n denotes the number of vertices in G. © 2006 Springer-Verlag.