The complexity of the node capacitated in-tree packing problem

In this paper, we deal with a node capacitated in-tree packing problem. The input consists of a directed graph, a root node, a node capacity function and edge consumption functions. The problem is to find the maximum number of rooted in-trees such that the total consumption of in-trees at each node does not exceed the capacity of the node. The problem is one of the network lifetime problems that are among the most important issues in the context of sensor networks. We reveal the computational complexity of the problem under various restrictions on consumption functions and graphs. For example, we consider general graphs, acyclic graphs and complete graphs embedded in the d-dimensional space R having edge consumption functions depending on distances between end nodes.