Presented by Yunho Kim Provable Software Lab, KAIST

Heavy-Tailed Phenomena in Satisfiabil-ity and Constraint Satisfaction Prob-

lemsby 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)


• 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)


• 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)


• Problem instance: quasigroup completion problem (N = 11, 30% pre-assignments)

The sample mean of the number of backtracks does diverge

Sam

ple

mea

n (n

umbe

r of b

ack-

track

s)

Number of runs

Introduction(4/4)


• 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


• Introduction




• Conclusion

Search Procedures


• 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)


• 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)


• 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


• 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


• Introduction




• Conclusion

Cumulative Distribution(1/2)


• 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)


• 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)


StandardExponential 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



• 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 FiniteVari-ance

Infinite Infi-nite



• 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)


• 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))



Log number of backtracks

Log(1-F(x)) ~ -®Log(x) + C’

Visual Check(2/3)


• Log-log plot of the cost distribution of the satisfi-able timetabling instance

• 1-F(x) = P{X > x} ~ Cx-®

Completion of timetablingLog(1-F(x))

Log number of backtracks

Visual Check(3/3)


• 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

SatzRelsat

Estimation of ®


• Calculated maximum likelihood estimates of ® using Hill estimator– k is sample size

• Since ® · 1, mean and variance are infinite

Unsatisfiable Instance


• 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


• Introduction




• Conclusion

Restarts


• 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


• Randomized rapid restarts(RRR) show better per-formance than deterministic

Contents


• Introduction




• Conclusion

Conclusion


• 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 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)


• Probability density function of Lévy distri-bution– Lévy have infinite mean and variance

Lévy Distribution(2/2)


• Cumulative distribution function of Lévy distribu-tion

Documents

Presented by Yunho Kim Provable Software Lab, KAIST