An Improved Restart Strategy for Randomized Backtrack Search Venkata P. Guddeti Constraint Systems Laboratory Universit

1. An Improved Restart Strategy for Randomized Backtrack Search Venkata P. Guddeti Constraint Systems Laboratory University of Nebraska-Lincoln Under the supervision of Dr. Berthe Y. Choueiry Guddeti: MS thesis defense

2. Outline • Summary of contributions • Background • Randomized BT search with restarts • Empirical evaluations • Conclusions & future research directions Guddeti: MS thesis defense

3. Summary of contributions • An improved restart strategy for randomized backtrack search (RDGR) • Evaluation & characterization • Comparison with BT, LS, ERA, RGR • Criterion: solution quality distribution • Problem types: GTAAP & random CSPs • As a result, we have identified • Regimes where a given technique dominates • Building blocks for designing cooperative, hybrid search Guddeti: MS thesis defense

4. Outline • Summary of contributions • Background • Constraint satisfaction problem (CSP) • Graduate Teaching Assistants Assignment Problem (GTAAP) • Search strategies: BT, LS, ERA • Randomized BT search with restarts • Empirical evaluations • Conclusions & future research directions Guddeti: MS thesis defense

5. CSP: Definition • Given P = (V, D, C): • V a set of variables • D a set of variable domains (values that a variable can take) • C a set of constraints • Objective: assign a value to each variable such that all constraints are satisfied In general, a CSP is NP-complete Guddeti: MS thesis defense

6. V1 V2 {d} ≠ ≠ ≠ V3 V4 {c, d, e, f} {a, b, c} {a, b, d} CSP: Representation • Variable → node • Domain → node label • Constraint → edge between nodes ≠ Guddeti: MS thesis defense

7. Context: GTAAP [Glaubius 01] Hiring & managing GTAs as instructors + graders • Given • A set of courses • A set of GTAs • A set of constraints that specify allowable assignments • Find a consistent & satisfactory assignment • Consistent: assignment breaks no (hard) constraints • Satisfactory: assignment maximizes • number of courses covered • happiness of the GTAs Guddeti: MS thesis defense

8. Constraint-based model • Variables (typically70 courses) • Grading, conducting lectures, labs & recitations • Values (30 GTAs) • Hired GTAs (+ preference for each value in domain) • Constraints • Unary, binary, global (e.g., capacity) • Objective • longest consistent solution (primary criterion) • maximize geometric mean of preferences (secondary criterion) Guddeti: MS thesis defense

9. V1 {d} V2 {c, d, e, f} ≠ ≠ ≠ V3 V4 {a, b, c} {a, b, d} Backtrack search (BT) Start with an empty assignment & expand it by instantiating one variable at a time ≠ Guddeti: MS thesis defense

10. BT (cont’d) • In theory, complete. In practice... forget it • Huge branching factor causes thrashing  backtrack never reaches early variables • Tested 12 ordering heuristics (Chap 3) • No significant difference • Use randomization & • restarts [Gomes et al. 98] Guddeti: MS thesis defense

11. Iterative-improvement search • Start with a complete assignment (=state), move to states that improve current one • Not complete • Tested: LS and ERA [Hui Zou, MS 2003] • Advantages: • Explores relatively wide portions of solution space • ERA solves tight instances, never solved before or since • Disadvantages • LS: local optimum & plateau cause stagnation • ERA: deadlock in over-constrained cases causes instability Guddeti: MS thesis defense

12. Outline • Summary of contributions • Background • Randomized BT search with restarts • Empirical evaluations • Conclusions & future research directions Guddeti: MS thesis defense

13. BT: Randomization & restarts 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 Guddeti: MS thesis defense

14. 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] • Bayesian approach [Kautz et al., 02] Guddeti: MS thesis defense

15. 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 Guddeti: MS thesis defense

16. RDGR • 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 Guddeti: MS thesis defense

17. Outline • Summary of contributions • Background • Randomized BT search with restarts • Empirical evaluations • Conclusions & future research directions Guddeti: MS thesis defense

18. Three main experiments • Effect of run time on RGR & RDGR • Choiceofr in RGR & RDGR • Relative performance of RDGR versus • Backtrack search (BT) [Glaubius 01] • Local Search (LS) [Zou 03] • Multi-Agent Search (ERA) [Liu et al. 02, Zou 03] • RGR All implementations use same platform and executed to the best of our abilities (internal competition) Guddeti: MS thesis defense

19. Evaluation criteria • Solution Quality Distribution (SQD) • cumulative distributions of solution quality • measured as percentage deviation from best known solution • Descriptive statistics • Mean, median, mode, std dev, max, min • 95% confidence interval using • Mann-Whitney U-test • Wilcoxon matched pairs signed-rank test Guddeti: MS thesis defense

20. Data sets • 8 real-world data sets (GTAAP) • 5 solvable, 3 over-constrained • Experiment repeated 500 times • 4 sets of randomly generated problems • Model B, 100 instances, each instance runs for 3 minutes Solvable <25,15,0.5,0.36> Unsolvable <25,15,0.5,0.36> <40,20,0.5,0.2> <40,20,0.5,0.5> Guddeti: MS thesis defense

21. 1. Effect of varying run time • RDGR consistently outperforms RGR • Running time does not affect the relative dominance Guddeti: MS thesis defense

22. 2. Choice ofrin RGR r = 1.1 for RGR for GTAAP & random CSPs Guddeti: MS thesis defense

23. 2. Choice of rin RDGR r = 1.1 for GTAAP r = 2 for random CSPs Guddeti: MS thesis defense

24. 3. Performance: SQDs • Under-constrained: ERA > RDGR > RGR > BT > LS • Over-constrained: RDGR > RGR > BT > LS > ERA Guddeti: MS thesis defense

25. 3. SQDs at phase transition • Solvable: ERA still wins for smallest deviations • Unsolvable: RDGR > RGR > BT > ERA > LS Guddeti: MS thesis defense

26. 3. Performance: RDGR vs. RGR • RDGR allows more restarts than RGR • RDGR is more stable than RGR Guddeti: MS thesis defense

27. Outline • Summary of contributions • Background • Randomized BT search with restarts • Empirical evaluations • Conclusions & future research directions Guddeti: MS thesis defense

28. Summary: algorithm dominance On GTAAP & randomly generated CSPs • Solvable instances ERA > RDGR > RGR > BT > LS • Over-constrained instances RDGR > RGR > BT > LS > ERA • At phase transition (statistically) RDGR > RGR > BT > ERA > LS (although ERA gives best results on solvable instances) Guddeti: MS thesis defense

29. Future research • Design ‘progress-aware’ restart strategies • Where cutoff value is changed during search • Design new search strategies • Hybrids: a solution from a given technique is fed to another • Cooperative: strategies applied where most appropriate within a given problem instance Guddeti: MS thesis defense

30. Thank you for your attention I welcome your questions.. Guddeti: MS thesis defense