We present the results of an empirical study of several constraint satisfaction search algorithms and heuristics. Using a random problem generator that allows us to create instances with given characteristics, we show how the relative performance of various search methods varies with the number of variables, the tightness of the constraints, and the sparseness of the constraint graph. A version of backjumping using a dynamic variable ordering heuristic is shown to be extremely effective on a wide range of problems. We conducted our experiments with problem instances drawn from the 50% satisfiable range.

The paper evaluates the effectiveness of learning for speeding up the solution of constraint satisfaction problems. It extends previous work (Dechter 1990) by introducing a new and powerful variant of learning and by presenting an extensive empirical study on much larger and more difficult problem instances. Our results show that learning can speed up backjumping when using either a fixed or dynamic variable ordering. However, the improvement with a dynamic variable ordering is not as great, and for some classes of problems learning is helpful only when a limit is placed on the size of new constraints learned.

In this work we study the problem of writing a Hermitian polynomial as a Hermitian sum of squares modulo a Hermitian ideal. We investigate a novel idea of Putinar-Scheiderer to obtain necessary matrix positivity conditions for Hermitian polynomials to be Hermitian sums of squares modulo Hermitian ideals. We show that the conditions are sufficient for a class of examples making a connection to the operator-valued Riesz-Fejer theorem and block Toeplitz forms. The work fits into the larger themes of Hermitian versions of Hilbert's 17-th problem and characterizations of positivity.