We consider the problem of allocating a large number of independent, equal-sized tasks to a heterogeneous computing platform. We use a nonoriented graph to model the platform, where resources can have different speeds of computation and communication. Because the number of tasks is large, we focus on the question of determining the optimal steady state scheduling strategy for each processor (the fraction of time spent computing and the fraction of time spent communicating with each neighbor). In contrast to minimizing the total execution time, which is NP-hard in most formulations, we show that finding the optimal steady state can be solved using a linear programming approach and, thus, in polynomial time. Our result holds for a quite general framework, allowing for cycles and multiple paths in the interconnection graph, and allowing for several masters. We also consider the simpler case where the platform is a tree. While this case can also be solved via linear programming, we show how to derive a closed-form formula to compute the optimal steady state, which gives rise to a bandwidth-centric scheduling strategy. The advantage of this approach is that it can directly support autonomous task scheduling based only on information local to each node; no global information is needed. Finally, we provide a theoretical comparison of the computing power of tree-based versus arbitrary platforms.