Structure Identification In Layered Precedence Networks

We consider the problem of identifying the structure of a network when precedence constraints are present. The associated graph is assumed to be directed, layered and with edges that exist only between successive layers. Such precedence networks arise in manufacturing processes resulting from a necessary or desirable sequencing of subassembly tasks to be completed in order to assemble the final product. Each node in the graph corresponds to a task and each edge represents a precedence constraint. We provide a simple and natural algorithm that can identify the network structure. We study the corresponding sample complexity as a function of the number of layers L, the maximum number of nodes, D, per layer and the maximum node in-degree M in the network. We show that, for networks that are start-time synchronized, i.e., networks in which all tasks per layer are coordinated to start simultaneously, identification with probability at least 1 - is an element of for some 0 < is an element of < 1 is guaranteed for a number of samples satisfying n = Omega(M log((L - 2)DM/is an element of)).