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Local Restartsin SAT Solvers

Vadim Ryvchin and Ofer Strichman

Technion, Haifa, Israel

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Restarts

SAT solvers use a "restart" policy• Following various criteria, the solver is forced to

backtrack to level 0.

Restarts have crucial impact on performance.

Motivation: avoid spending too much time in ‘bad’ branches:• no easy-to-find satisfying assignment • no opportunity for fast learning of strong clauses.

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Restarts Motivation Best time to restart:

• When algorithm spends too much time under a “wrong” branch

No solution

Perform restart

solution

decision level kdecision level k

decision level 1decision level 1

decision level 0decision level 0

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The basic measure for restarts

All existing techniques use the number of conflicts learned as of the previous restart.

The difference is only in the method of calculating the threshold.

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Restarts strategies

1. Arithmetic (or fixed) series. • Parameters: x, y. • Init(t) = x• Next(t)=t+y

Used in: • Berkmin (550, 0)• Eureka (2000, 0)• Zchaff 2004 (700, 0)• Siege (16000, 0)

Arithm(2000, 0) , Arithm(1000, 10)

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Restart NumberT

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Restarts Strategies (cont.)

2. Geometric series.• Parameters: x, y. • Init(t) = x• Next(t)=t*y

Used in• Minisat 2007 (100, 1.5)

Geo(100, 1.5)

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Restart NumberTh

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Restarts Strategies (cont.)

3. Inner-Outer Geometric series. • Parameters: x, y, z. • Init(t) = x• if (t*y < z)

- Next(t) = t*y• else

- Next(t) = x- Next(z) = z*y

Used in:• Picosat (100, 1.1, 1000)

Inner-Outer (100, 1.1, 100)

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1 17 33 49 65 81 97 113 129 145 161 177 193

Restart NumberTh

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Restarts Strategies (cont.)

4. Luby et al. series. • Parameter: x.• Init(t) = x

• Next(t) = ti*x

Used in:• RSat (512)• Tinisat (512)

Luby(512)

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1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97

Restart NumberTh

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ti =1 1 2 1 1 2 4 1 1 2 1 1 2 4 8 1 1 2 1 1 2 4 1 1 2 1 1 2 4 8 16 1 1 2 1 1 2 4 1 1 2 1 1 2 4 8 …

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Global vs. Local

Recall: all techniques measure the number of conflicts as of the previous restart.

This is a global criterion.

Somewhat disregards the original motivation of focusing on ‘bad’ branches.

Local criteria: measure difficulty of branches.

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Global Restarts

Perform restart

Explored sub tree

No solution solutionNo solution solution

decision level kdecision level k

decision level 1decision level 1

decision level 0decision level 0

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Local Restarts

Perform restart

Explored sub tree

No solution solution

decision level k

decision level 1

decision level 0

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Calculating the number of conflicts under a branch

We worked with the following local measure:• Number of conflicts above each level.

For each decision level d maintain a counter c(d): • When making a decision at level d,

- c(d) = #global conflicts.• When backtracking to level d:

- res = #global conflicts - c(d).• If res > threshold, restart.

All the presented strategies can be local.

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Dynamic Criteria

A new dimension for the restart strategy: the decision level d. • The restart threshold is a function of d.

Higher decision level smaller threshold.

Each of the local strategies can be made dynamic.

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Dynamic(1000, 10, 2, 50) - Current x =1000

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1 79 157 235 313 391 469 547 625 703 781 859 937

Decision Level

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Dynamic Criteria

Dynamic-fix. • d – decision level, • i – iteration • Parameters:

x, y, c.

• Init(t) = x• Next(t) = x + y * i – (c * d)

min

Constant

Need a minimum !

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Dynamic(1000, 10, 2, 50) - Current x =1000

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1 79 157 235 313 391 469 547 625 703 781 859 937

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Dynamic Criteria

Dynamic-fix. • d – decision level, • i – iteration • Parameters:

x, y, c, min > 0.

• Init(t) = x• Next(t) = max(x + y * i – (c * d), min)

min

Constant

Not a restart number

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Experimental Results

Solvers: Minisat 2007 and Eureka Benchmarks: 100 instances from SAT-race 2006

(the TS1,TS2 test sets). Timeout: 30 minutes.

• Instances that timed-out are included and contribute 30 minutes

Restart configurations: 40• Chosen dynamically following ‘hill climbing’• Looked also for best parameters for global restarts

Experimental Results - Minisat

Restart Strategy

Local Global Speedup

Arithmetic 7.93 9.52 x1.16

Geometric 9.55 10.64 x1.11

Inner-Outer 7.37 8.86 x1.2

Luby 7.89 8.98 x1.13

Original vs. Best Local

7.37 10.64 x1.44

Experimental Results (fails) – Minisat

Restart Strategy

Local Global improvement

Arithmetic 10 10 1

Geometric 11 14 x1.27

Inner-Outer 8 10 x1.25

Luby 9 11 x1.22

Original vs. Best Local

8 14 x1.75

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Dynamic Restart Unsat improvement:

• x1.19 speedup (4.09h vs. 4.90h)• x1.75 in unsolved instances (4 vs. 7).

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Dynamic Restart Unsat improvement:

• x1.19 speedup (4.09h vs. 4.90h)• x1.75 in unsolved instances (4 vs. 7).

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Conclusions

Conclusion: • Making restarts local helps all restarts strategies.• Improves average run time

- ~ x1.44 in Minisat- ~ x1.17 in Eureka

• Improves fails (timeout = 30 min) by - ~ x1.75 in Minisat- ~ x1.27 in Eureka