Backtracking Deterministic Annealing for Constraint Satisfaction Problems

We present a new deterministic annealing approach to the solution of quadratic constraint satisfaction problems with complex interlocking constraints, such as exemplified in polyomino tiling puzzles. We first analyze the dynamical properties of the solution strategies implemented by deterministic annealing (DA) in the analog neural representation of Potts-MeanField (PMF) and penalty-function-based competitive layer model (CLM) neural networks, revealing a similar mechanism. The key idea of our extension of these plain DA approaches is motivated by classical backtracking algorithms. We show that their ability for iterative local pruning of the search space can be implemented within the framework of DA by introducing local temperature parameters which are “reheated” when locally unresolved conflicts occur. To achieve the pruning of the search space, reheating is accompanied by a modification of the constraint-implementing weight matrix to reduce the chance of reentering the same local configuration. The weight changes provide a learning mechanism that facilitates the generation of a solution for subsequent runs. We demonstrate the benefits of the resulting “backtracking deterministic annealing” algorithm (BDA) by applying it to a pentomino tiling problem. We show that the method reliably finds perfect solutions to the task, while the plain DA approach for both PMF and CLM is unable to solve the task in a comparable or even considerably larger number of iterations.