Foundations of Constraint Processing CSCE421/821, Spring 2008:

1. Extensions of BT Search Foundations of Constraint Processing CSCE421/821, Spring 2008: www.cse.unl.edu/~choueiry/S08-421-821/ Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 123B choueiry@cse.unl.edu Tel: +1(402)472-5444 More on BT search

2. Outline • Variations of backtrack search • Backtrack search for optimization More on BT search

3. Variations on BT search • Bounded number of backtracks search • Bounded backtrack-depth search • Limited discrepancy search (LDS) • 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) More on BT search

4. Credit-Based Search • Start with a given credit (usually n3) • 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 • ECLiPSe uses it in conjunction with backtrack-bounded search More on BT search

5. Randomized BT search In systematic backtrack 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 More on BT search

6. 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] More on BT search

7. 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 More on BT search

8. 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 More on BT search

9. Outline • Variations of backtrack search • Backtrack search for optimization Courtesy of Markus Fromherz • Branch and bound • Binary search • Iterative deepening More on BT search

10. 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 More on BT search

11. 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 More on BT search

12. 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 More on BT search

13. Iterative deepening • Restart search with an increasing upper limit on the solution quality until a solution is found More on BT search