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 dont 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