1. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search1 Foundations of Constraint Processing CSCE421/821, Fall 2004: www.cse.unl.edu/~choueiry/F04-421-821/ Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 123B choueiry@cse.unl.edu Tel: +1(402)472-5444 More on BT Search

2. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search2 Outline Lookahead Variations of backtrack search Backtrack search for optimization

3. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search3 Lookahead Rationale –As decisions are made (conditioning) eliminate inconsistent choices in future sub-problem –Domain annihilation of a future variable avoids expansion of useless portions of the tree Techniques –Partial: forward-checking (FC), directional arc-consistency (DAC) –Full: Maintaining arc-consistency (MAC) MAC: more pruning at the cost of more consistency checks Empirical results: LooseTight SparseFCMAC DenseFC

4. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search4 Outline Lookahead Variations of backtrack search Backtrack search for optimization

5. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search5 Variations on BT search Bounded number of backtracks search Bounded backtrack-depth search Limited discrepancy search –Heuristic may be blind at shallowest level of search-tree –Disobey heuristic a given number of times Credit-based backtrack search Randomized backtrack search (+ restart)

6. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search6 Credit-Based Search Start with a given credit (usually n 3 ) Assign ½ credit to current assignment, ½ to the remaining ones Keep going, in a depth-first manner until credit is used up, (chronoligically) backtrack from there ECL i PS e uses it in conjunction with backtrack-bounded search

7. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search7 Randomized BT search Ordering of variables/values determines which parts of the solution space are explored –Randomization allows us to explore wider portion of search tree Thrashing causes stagnation of BT search –Interrupt search, then restart In systematic backtrack search

8. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search8 Restart strategies Fixed-cutoff & universal strategy [Luby et al., 93] Randomization & Rapid restarts (RRR) [Gomes et al., 98] –Fixed optimal cutoff value –Priori knowledge of cost distribution required Randomization & geometric restarts (RGR) [Walsh 99] Randomization & dynamic geometric restarts (RDGR) [Guddeti 04] Bayesian approach [Kautz et al., 02]

9. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search9 RGR [Walsh 99] Static restart strategy As the cutoff value increases, RGR degenerates into randomized BT –Ensures completeness (utopian in our setting) –But… restart is obstructed –… and thrashing reappears  diminishing the probability of finding a solution

10. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search10 RDGR [Guddeti 04] Randomization & Dynamic Geometric Restarts Cutoff value –Depends on the progress of search –Never decreases, may stagnate –Increases at a much slower rate than RGR Feature: restart is ‘less’ obstructed

11. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search11 Outline Lookahead Variations of backtrack search Backtrack search for optimization

12. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search12 BT search for optimization Courtesy of Markus Fromherz Branch & bound –Application to over-constrained CSPs Binary search Iterative deepening Etc.

13. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search13 Branch & Bound Branch & bound –Find a first solution, compute its quality, call it the incumbent –Search for other solutions, comparing them with the incumbent –As soon a better solution is found, make it the incumbent –Continue until you run out of time, patience, or solutions

14. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search14 B&B: over-constrained-CSPs Max-CSP –Goal: minimize the number of broken constraints (while instantiating all variables) Maximize solution length –Goal: maximize number of variables instantiated (while satisfying all constraints) We compare the incumbent and the partial solution along current path according to –the number of broken constraints or –the number of instantiated variables

15. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search15 Binary search Given l, u lower and upper bounds of the quality of the solution Check whether there is a solution in [l, u+l/2]=[u,l’] –If there is, set the bounds [u, u+l’/2] and search for a solution –If there is not, set the bounds to [l’, l+l’/2] and search for a solution… Restart search with progressively narrower lower and upper bounds on the solution

16. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search16 Iterative deepening Restart search with an increasing upper limit on the solution quality until a solution is found

17. Foundations of Constraint Processing, Fall 2004 November 18, 2004More on BT search17 Rest of the course? Local search Binary vs. non-binary Phase Transition All-diff constraint (Shasha?) Backtrack-free, backtrack-bounded search Temporal CSPs Interchangeability Dynamic CSPs