Abstract: "We consider two variants of the task assignment problem for distributed systems. The first is the problem of finding a minimum cost assignment when one of the processors has a limited memory. The second is the problem of finding an assignment that minimizes the maximum processor load. Both problems are NP-hard, even if the communication graph is a tree. We present exact algorithms and approximation schemes for these problems for the case where the communication graph is a partial k-tree. Faster algorithms are presented for the case of trees with uniform costs. We also show that, if the communication graph is unrestricted, there is no fully polynomial-time approximation scheme for the memory-constrained problem unless P = NP."