Henrik Jönsson and Bo Söderberg Deterministic Annealing and Non-Linear Assignment Abstract: For combinatorial optimization problems that can be formulated as Ising or Potts spin systems, the Mean Field (MF) approximation yields a versatile and simple ANN heuristic, Deterministic Annealing. For problems involving assignments (or permutations), the situation is more complex -- the natural analog of the MF approximation lacks the simplicity present in the Potts and Ising cases. In this article the difficulties associated with this issue are investigated, and the options for solving them discussed. Improvements to existing Potts-based MF-inspired heuristics are suggested, and the possibilities for defining a proper variational approach are scrutinized. LU TP 01-16