Heuristics for backtracking algorithms

1. Heuristics for backtracking algorithms • Variable ordering (important) • what variable to branch on next • Value ordering (not yet so important) • given a choice of variable, what order to try values • Constraint ordering (not so important) • what order to propagate constraints • most likely to fail or cheapest propagated first

2. Variable ordering • Domain dependent heuristics • Domain independent heuristics • Static variable ordering • fixed before search starts • Dynamic variable ordering • chosen during search

3. Variable ordering: Possible goals • Minimize the underlying search space • static ordering example: Suppose x1, x2, x3, x4 with domain sizes 2, 4, 8, 16. Compare static ordering x1, x2, x3, x4 vs x4, x3, x2, x1 • Minimize expected depth of any branch • Minimize expected number of branches • Minimize size of search space explored by backtracking algorithm • intractable to find “best” variable

4. Basic idea • Assign a heuristic value to a variable that estimates how difficult it is to find a satisfying value for that variable • Principle: most likely to fail first • or don’t postpone the hard part

5. Some variable ordering heuristics • minimum domain size (dom) • maximum degree (deg) • most constraining • plus: combining — tie break strategies • e.g., dom + deg, dom / deg

6. Open questions • Dynamic variable ordering heuristics for non-binary CSPs

7. Value ordering • All solutions • value ordering not so important • One solution • if a solution exists, there exists a perfect value ordering

8. Value ordering: Intuition • Goal: minimize size of search space explored • Principle: • given that we have already chosen the next variable to instantiate, choose first the values that are most likely to succeed