The computational complexity and the approximation algorithms of the optimal p-source communication spanning tree (p-OCT) problem were investigated. Let G be an undirected graph with nonnegative edge lengths. Given p vertices as sources and all vertices as destinations, and also given arbitrary requirements between sources and destinations, we investigated the problem how to construct a spanning tree of G such that the total communication cost from sources to destinations is minimum, where the communication cost from a source to a destination is the path length multiplied by their requirement. For any fixed integer p⩾2, we showed that the problem is NP-hard even for metric graphs. For metric graphs of n vertices, we show a 2-approximation algorithm with time complexity O(np−1). For general graphs, we present a 3-approximation algorithm for the case of two sources.
