1. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense1 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

2. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense2 Outline Summary of contributions Background Randomized BT search with restarts Empirical evaluations Conclusions & future research directions

3. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense3 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

4. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense4 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

5. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense5 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

6. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense6 CSP: Representation Variable → node Domain → node label Constraint → edge between nodes ≠ ≠ V3V3 V4V4 V2V2 V1V1 ≠ ≠ {d}{c, d, e, f} {a, b, c}{a, b, d}

7. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense7 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 1.number of courses covered 2.happiness of the GTAs

8. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense8 Constraint-based model Variables (typically 70 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)

9. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense9 Backtrack search (BT) Start with an empty assignment & expand it by instantiating one variable at a time ≠ ≠ V3V3 V4V4 V2V2 V1V1 ≠ ≠ {d}{c, d, e, f} {a, b, c}{a, b, d}

10. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense10 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]

11. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense11 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

12. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense12 Outline Summary of contributions Background Randomized BT search with restarts Empirical evaluations Conclusions & future research directions

13. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense13 BT: Randomization & restarts 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

14. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense14 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]

15. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense15 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

16. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense16 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

17. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense17 Outline Summary of contributions Background Randomized BT search with restarts Empirical evaluations Conclusions & future research directions

18. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense18 Three main experiments 1.Effect of run time on RGR & RDGR 2.Choice of r in RGR & RDGR 3.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)

19. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense19 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

20. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense20 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 Unsolvable

21. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense21 1. Effect of varying run time RDGR consistently outperforms RGR Running time does not affect the relative dominance

22. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense22 2. Choice of r in RGR r = 1.1 for RGR for GTAAP & random CSPs

23. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense23 2. Choice of r in RDGR r = 1.1 for GTAAP r = 2 for random CSPs

24. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense24 3. Performance: SQDs Under-constrained: ERA > RDGR > RGR > BT > LS Over-constrained: RDGR > RGR > BT > LS > ERA

25. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense25 3. SQDs at phase transition Solvable: ERA still wins for smallest deviations Unsolvable: RDGR > RGR > BT > ERA > LS

26. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense26 3. Performance: RDGR vs. RGR RDGR allows more restarts than RGR RDGR is more stable than RGR Data sets123456 Average restarts RGR16.717.422.514.722.419.5 RDGR74.559.9167.439.1 46.2 Data sets123456 Standard deviation RGR2.81.10.71.0 1.2 RDGR0.70.80.60.90.71.1

27. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense27 Outline Summary of contributions Background Randomized BT search with restarts Empirical evaluations Conclusions & future research directions

28. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense28 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)

29. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense29 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

30. Constraint Systems Laboratory Oct 21, 2004Guddeti: MS thesis defense30 Thank you for your attention I welcome your questions..