In this paper, we consider the problem of allocating a large number of independent, equal-sized tasks to a heterogeneous grid computing platform. We use a non-oriented graph to model a grid, where resources can have different speeds of computation and communication, as well as different overlap capabilities. We show how to determine 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). This result holds for a quite general framework, allowing for cycles and multiple paths in the interconnection graph, and allowing for several masters. Because spanning trees are easier to deal with in practice (there is a single path from the master to each node), a natural question arises: how to extract the best spanning tree, i.e. the one with optimal steady-state throughput, out of a general interconnection graph? We show that this problem is NP-hard. Even worse, we show that there exist heterogeneous networks for which the optimal spanning tree has a throughput which is arbitrarily bad in front of the throughput that can be achieved by the optimal (multiple-path) solution. Still, we introduce and compare several low-complexity heuristics to determine a sub-optimal spanning tree. Fortunately, we observe that the best heuristics do achieve an excellent performance in most experiments.