Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems by Carla P. Gomes,...
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Transcript of Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems by Carla P. Gomes,...
Heavy-Tailed Phenomena in Satisfiabil-ity and Constraint Satisfaction Prob-
lems
by Carla P. Gomes, Bart Selman, Nuno Crato and henry Kautz
Presented by Yunho KimProvable Software Lab, KAIST
Contents
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
2/28
• Introduction
• Search procedures and problem do-mains
• Cost distributions of backtrack search
• Consequences for Algorithm Design
• Conclusion
Introduction(1/4)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
3/28
• The DPLL algorithm is a complete algorithm for deciding the satisfiability of propositional logic formulas– It is guaranteed that eventually either the DPLL algo-
rithm finds a satisfying model or proves the formula is unsatisfiable
The iterative version of DPLL algorithm 1 status = preprocess(); 2 if (status!=UNKNOWN) return status; 3 while(1){ 4 decide_next_branch(); 5 while(1){ 6 status = deduce(); 7 if (status == CONFLICT){ 8 blevel = analyze_conflict(); 9 if (blevel == 0) 10 return UNSAT; 11 else backtrack(blevel); 12 } 13 else if (status == SAT) 14 return SAT; 15 else break; 16 } 17 }
Introduction(2/4)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
4/28
• At each step a heuristic is used to select the next branch variable– A branch heuristic scores each variable in some manner
and select the highest one
• Randomization can be used for tie-breaking– If several choices are ranked equally, choose among
them at random– All variables that receive scores within H-percent of the
highest score are considered equally good• H is a heuristic equivalence parameter
• Empirically it is known that randomized branching heuristics is effective on hard instances
Introduction(3/4)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
5/28
• Problem instance: quasigroup completion problem (N = 11, 30% pre-assignments)
The sample mean of the number of backtracks does diverge
Sam
ple
mean
(n
um
ber
of
back
-tr
ack
s)
Number of runs
Introduction(4/4)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
6/28
• The authors have observed the erratic behavior of the mean and the variance of the search cost dis-tributions on a same instance
• The mean cost calculated over an increasing number of runs, on the same satisfiable problem instance, of a randomized backtrack search pro-cedure does diverge.
• The authors have not found unsatisfiable in-stances with heavy-tailed behavior
Contents
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
7/28
• Introduction
• Search procedures and problem do-mains
• Cost distributions of backtrack search
• Consequences for Algorithm Design
• Conclusion
Search Procedures
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
8/28
• The authors modified two state-of-the-art SAT solvers(at that time), Satz and Relsat
• Both solvers hire similar occurrences-based deci-sion heuristics
• Satz employs chronological backtracking while Relsat uses non-chronological backtracking
Quasigroup Completion Problem(1/2)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
9/28
• A quasigroup is an ordered pair(Q, ¢), where Q is a set and (¢) is a binary operation on Q such that the equations a ¢ x = b and y ¢ a = b are uniquely solvable for every pair of el-ements a, b in Q
• The order N of the quasigroup is the cardinality of the set Q
• The multiplication table of a finite quasigroup is a Latin square– An N £ N table filled with n different symbols in such a way that each
symbol occurs exactly once in each row and exactly once in each col-umn
Order 4 quasigroup Order 10 quasi-group
Quasigroup Completion Problem(2/2)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
10/28
• The quasigroup completion problem – determining whether the remaining entries of the partial
Latin square can be filled in such a way that we obtain a complete Latin square
• The quasigroup completion problem is NP-com-plete
32% pre-as-signed
Other Problems
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
11/28
• Timetabling, planning and instances in the Di-macs Challenge benchmark are also considered
• Timetabling problem is to determine whether there exists a feasible schedule that consider a set of pairing and distribution constraints
• Planning is to find a sequence of actions that transform an initial state to a goal state
Contents
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
12/28
• Introduction
• Search procedures and problem do-mains
• Cost distributions of backtrack search
• Consequences for Algorithm Design
• Conclusion
Cumulative Distribution(1/2)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
13/28
• Data was produced by running the randomized backtrack search procedure 10,000 times on the same instance
• Even though 50% of the runs solve the instance in 1 backtrack or less, after 100,000 backtracks 0.5% of the runs were still not completed
Number of backtracks
Cumulative fraction of suc-cessful runs
Completion of quasi-group
Cumulative Distribution(2/2)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
14/28
• A solution is found in 1,000 backtracks or less in 80% of runs
• However, 5% of the runs do not result in a solu-tion even after 1,000,000 backtracks
Number of backtracks
Cumulative fraction of suc-cessful runs
Timetabling
Heavy-Tailed Distributions(1/3)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
15/28
Standard
Exponential decay
e.g. Normal:
P{X>x} ~ Ce-x2
for some C > 0
Heavy-Tailed
Power law decay
e.g. Pareto-Levy:
P{X>x} ~ Cx-®
where for some 0 < ® < 2 and C > 0
Power Law Decay
Standard Distribution(finite mean & variance)
Exponential Decay
Heavy-Tailed Distributions(2/3)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
16/28
• We consider distributions that asymptotically have “heavy tails”, namely,
where for some 0 < ® < 2 and C > 0
• The ® is referred to as the index of stability of the distribu-tion– The lower the index, the heavier the tail
• Heavy-tailed distributions have finite/infinite mean and infi-nite variance
P{X>x} ~ Cx-®
0 < ® · 1 1 < ®
Mean Infinite Finite
Vari-ance
Infinite Infi-nite
Heavy-Tailed Distributions(3/3)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
17/28
• Comparison of tail probabilities P{X > c} – Cauchy distribution is the heavy-tailed distribution which has
® = 1.0– Levy distribution is the heavy-tailed-distribution which has ® =
0.5
Visual Check(1/3)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
18/28
• Log-log plot of the cost distribution of the satisfi-able completion of quasigroups instances
• 1-F(x) = P{X > x} ~ Cx-®
N = 15, 40% pre-assignments
Completion of quasigroups
Log(1-F(x))
N = 15, 30% pre-assignments
N = 11, 30% pre-assignments
Log number of backtracks
Log(1-F(x)) ~ -®Log(x) + C’
Visual Check(2/3)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
19/28
• Log-log plot of the cost distribution of the satisfi-able timetabling instance
• 1-F(x) = P{X > x} ~ Cx-®
Completion of timetabling
Log(1-F(x))
Log number of backtracks
Visual Check(3/3)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
20/28
• Log-log plot of the cost distribution of the satisfi-able logistics planning from two different SAT solvers
• 1-F(x) = P{X > x} ~ Cx-®
Logistics planning
Log(1-F(x))
Log number of back-tracks
Satz
Relsat
Estimation of ®
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
21/28
• Calculated maximum likelihood estimates of ® using Hill estimator– k is sample size
• Since ® · 1, mean and variance are infinite
Unsatisfiable Instance
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
22/28
• Log-log plot of the cost distribution of unsatisfi-able completion of quasigroups instnces
• 1-F(x) = P{X > x} ~ Cx-®
Completion of quasigroups
Log(1-F(x))
Log number of back-tracks
Contents
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
23/28
• Introduction
• Search procedures and problem do-mains
• Cost distributions of backtrack search
• Consequences for Algorithm Design
• Conclusion
Restarts
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
24/28
• Restart after a fixed number of backtracks pre-vent a solver from entering pitfall
Total number of backtracks
Effect of restarts on a quasigroup in-stanceN = 20, 5% pre-assignments
Log(1-F(x))
No restarts
With restarts
Without restarts and given a total of 300 backtracks, 70% of runs failed
With restarts, only 0.01% of runs failed
Restarts
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
25/28
• Randomized rapid restarts(RRR) show better per-formance than deterministic
Contents
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
26/28
• Introduction
• Search procedures and problem do-mains
• Cost distributions of backtrack search
• Consequences for Algorithm Design
• Conclusion
Conclusion
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
27/28
• The authors show the suitability of heavy-tailed distributions in modeling the runtime behavior of DPLL SAT solver with the random decision heuris-tic
• Restarts can exploit the mass of probability on the left of the cost distributions
Reference
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
28/28
• Heavy-Tailed Phenomena in Satisfiability and Constraint Sat-isfaction Problemsby Carla P. Gomes, Bart Selman, Nuno Crato and Henry Kautzin Journal of Automated Reasoning 24: 67-100, 2000
Lévy Distribution(1/2)
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems Yunho Kim, Provable Software Lab, KAIST
29/28
• Probability density function of Lévy distri-bution– Lévy have infinite mean and variance