Minimal forward checking with backmarking and conflict-directed backjumping

Dent and Mercer (1996) have introduced an algorithm called minimal forward checking (MFC) which always performs no worse than forward checking (FC) in terms of number of compatibility checks and node expanded given the same variable and value orderings. In this paper we describe an algorithm which extends MFC with backmarking and conflict-directed backtracking. The new algorithm has a smaller space complexity than MFC. Experiments were conducted to compare it with MFC and some regular FC based algorithms. The results show that the new algorithm always performs at least as good as its "non-lazy" counterpart. It outperforms MFC on average and its edge over MFC is particularly clear for problems near to phase transitions. Interestingly, the minimum width variable ordering heuristic appears to be a better choice than the fail-first heuristic for the new algorithm in many occasions, particularly for sparsely constrained problems.