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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. Outline. Summary of contributions Background Randomized BT search with restarts
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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
Outline • Summary of contributions • Background • Randomized BT search with restarts • Empirical evaluations • Conclusions & future research directions Guddeti: MS thesis defense
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
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
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
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
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
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
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
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
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
Outline • Summary of contributions • Background • Randomized BT search with restarts • Empirical evaluations • Conclusions & future research directions Guddeti: MS thesis defense
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
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
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
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
Outline • Summary of contributions • Background • Randomized BT search with restarts • Empirical evaluations • Conclusions & future research directions Guddeti: MS thesis defense
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
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
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
1. Effect of varying run time • RDGR consistently outperforms RGR • Running time does not affect the relative dominance Guddeti: MS thesis defense
2. Choice ofrin RGR r = 1.1 for RGR for GTAAP & random CSPs Guddeti: MS thesis defense
2. Choice of rin RDGR r = 1.1 for GTAAP r = 2 for random CSPs Guddeti: MS thesis defense
3. Performance: SQDs • Under-constrained: ERA > RDGR > RGR > BT > LS • Over-constrained: RDGR > RGR > BT > LS > ERA Guddeti: MS thesis defense
3. SQDs at phase transition • Solvable: ERA still wins for smallest deviations • Unsolvable: RDGR > RGR > BT > ERA > LS Guddeti: MS thesis defense
3. Performance: RDGR vs. RGR • RDGR allows more restarts than RGR • RDGR is more stable than RGR Guddeti: MS thesis defense
Outline • Summary of contributions • Background • Randomized BT search with restarts • Empirical evaluations • Conclusions & future research directions Guddeti: MS thesis defense
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
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
Thank you for your attention I welcome your questions.. Guddeti: MS thesis defense