Reconciling - Universität des Saarlandesraphael/bachelor/downloads/talk4...separator entry denotes branching level Depth-ﬁrst search explores a single path in a search tree, represented

Graduate Seminar 2008

ReconcilingCopying and Trailing for Constraint Programming Systems

Raphael Reischuk

Graduate Seminar PS LabOctober 13, 2008

UN I

V E R S IT AS

SA

R A V I E N

SI S

Advisor: Guido TackSupervisor: Gert Smolka

Raphael Reischuk Graduate Seminar 2008

Overview

• motivation

• how to solve constraint problems?

• some observations

• a hybrid architecture

• reconciling copying and trailing

• experimental results

• future work


Motivation

• Copying based CP solvers are efficientfor integer and set variables (FD)not for SAT problems

• Trailing based SAT solvers are efficientfor Boolean variablesnot for integer and set variables

• No combination of both systems so far

• New: hybrid solver for hybrid problems

Gecode

MiniSat

hybridsolver

branching

x=5 x!5

y


How to solve problems?

CSP

integer and set variables

Boolean variables

solveusing

Gecode

solve using SATtechniques

Gecode propagators

clause propagators

• two independent components ?➟ solve them separately, concurrently


How to solve problems?

CSP

integer and set variables

Boolean variables

solveusing

Gecode

solve using SATtechniques

Gecode propagators

clause propagators

• two independent components ?➟ solve them separately, concurrently

• more interesting: hybrid problemswith shared variables, shared propagators, shared dependencies

solveusinghybridsolverhybrid

propagators


Observations

• solvers differ in

• how information is maintained(local vs. global information)

• restoration techniques(copying vs. trailing)


Maintaining Information

local information(copying)

global information(trailing)

propagatorvariabledependencies


subsumption

domain

domain‘

subsumption


domain

time


domain‘‘

+ subsumption

+ watched literals+ garbage collection

copy

copy

propagation

state of the solver

state of the solver

state of the solver

Trail

x≠5

y≠7

z≠4b≠1

x≠3

+ propagator rewriting

state of a solver

local part global part

collected ina space

several clones of a spacein memory

available at a single persistent

location

never copied

replace actual space by previous one

undo the modifications successively

storage:

restoration:

information:


Information in a Hybrid Solver

A hybrid solver has to deal with local and global information.

recomputation stack

(0, desc)

(1, desc)

(1, desc)

(0, desc)

(clone)

0

1

2

3

4

(clone)

(clone)

trailing stack

v17 ! 0

v52 ! 1

...

v12 ! 1

v37 ! 0

...

v13 ! 8

v13 ! 5

...

v23 ! 0

v67 ! 1

...

3

0

1

2

4

hybrid stack

v17 ! 0

v52 ! 1

...

v12 ! 1

v37 ! 0

...

v13 ! 8

v13 ! 5

...

v23 ! 0

v67 ! 1

...

3

0

1

2

4

branchingsmodifications

(0, desc) (clone)

(1, desc)

(1, desc) (clone)

(0, desc)

(clone)

fixed points


Architecture: DFS Stack

information in separator entries

information between separator entries

separator entrydenotes branching level

Depth-first search explores a single path in a search tree, represented by a stack.

We need to have new propagator types, new dependency types and new variable types.

state of the solver

state of the solver


Restoration Techniques

local information(recomputation)

global information(undoing changes)



domain

domain‘


domainrestoration


domain‘‘

jump to previous

state

modify statein-place

(„move upwards“)

copy

copy

Trail

x≠5

y≠7

z≠4b≠1

x≠3

failure

untrail

last clone

hybrid stack

......

......

(0, desc) (clone)

(0, desc)

(1, desc) (clone)

(1, desc)

(clone)

(clone)

untrail

sr

sf


Restoration: Full Copying

Simplest restoration technique: full copying

restoration level

failure level

untrail(1)

commits(4)

failure

schedule(3)

hybrid stack

......

......

(0, desc) (clone)

(0, desc)

(0, desc)

(1, desc)untrail(1)

(1, desc)

...

commits(4)

s!

sr

sf


Restoration: Full Recomputation

More involved restoration technique: full recomputation

restoration level

failure level

level of last clone

1) untrail (global part)

2) fetch clone from memory (local part)

3) schedule copied propagators depending on trailed variables explicitly (local part)

4) recompute (local part)

untrail(1)

commits(4)

failure

schedule(3)

hybrid stack

......

......

(0, desc) (clone)

(0, desc)

(0, desc)

(1, desc)untrail(1)

(1, desc)

...

commits(4)

s!

sr

sf


Restoration: Full Recomputation

More involved restoration technique: full recomputation

restoration level

failure level

level of last clone

1) untrail (global part)

2) fetch clone from memory (local part)

3) schedule copied propagators depending on trailed variables explicitly (local part)

4) recompute (local part)

x = 3⇔ ¬bwhy scheduling?

• constraint• copied variable x has domain

{1,2,3} in last clone• trailed variable b still set to 0• recomputation adds

branching constraints and computes one fixed point

• variable x is never modified since propagator is not scheduled

x = 3⇔ ¬b

x = 3⇔ ¬b

untrail(1)

commits(4)

failure

schedule(3)

compute fixed point (5)and place copy (6)

commits withfixed point computation

(7)

hybrid stack

......

......

(0, desc) (clone)

(0, desc)

(0, desc)

(1, desc)untrail

(1)

(0, desc)

...

commits(4)

s!

sr

sf

sm


Restoration: Adaptive Recomputation

Even more involved restoration technique: adaptive recomputation

restoration level

failure level

level of last clone 1. untrail (global part)

2. fetch clone from memory (local part)

3. schedule copied propagators depending on trailed variables explicitly (local part)

4. recompute (local part)

5. compute fixed point (both parts)

6. place copy (local part)

7. recompute and compute fixed point in each level (both parts)


Experimental Results 1: Overhead

+10.8 % +0.6 %

+6.3 %

+6.8 %

+12.8 % +12.6 %

Runtime overhead for non-trailed variables

Gecode Overhead


Experimental Results 2: Overhead

+76%

+69%

Runtime overhead for trailed variables with copied propagators

Gecode Overhead

+2%

+14%

+28%+48%

+29%


Experimental Results 3: Gain

Runtime

Gecode Hybrid Solver

Trailed variables with shared propagators

Propagation Steps (·106)

-31% -29%

-32%

-69%

+11% +95%

+105%

-37%

➟ Concerning runtime, we clearly outperform Gecode for this class of problems.Additionally, our hybrid solver uses only about one forth of the memory Gecode uses.



Runtime


Trailed variables with shared propagators using clause learning


-99,996%-99,4% -99,8%

-95,3% -99,995% -98,7%-99,4%

-95,0%

➟ Concerning runtime, we dramatically outperform Gecode.Because of the learned clauses, our solver uses more memory than Gecode.

40s ➟ 1,8ms1.156s ➟ 54s



Runtime


Trailed variables with shared propagators using clause learning


-99,996%-99,4% -99,8%

-95,3% -99,995% -98,7%-99,4%

-95,0%

➟ Concerning runtime, we dramatically outperform Gecode.Because of the learned clauses, our solver uses more memory than Gecode.

+44

+606

+695

+31.148

60v 600v 718v 78v

less propagationwith morepropagators

Propagators


Conclusions

• Our hybrid solver integrates trailing in the copying based solver Gecode. It reconciles two restoration techniques.

• The hybrid solver outperforms copying based solvers by using trailing for certain variables and SAT techniques on top.

• With the implementation of new shared propagators, our hybrid solver may solve more problems efficiently.


Future Work

• For pure SAT problems however, MiniSat [9] is much faster. There are still many SAT techniques to implement. [3,17,20]

• Another way of reconciling copying solvers with SAT solvers is the lazy clause generation approach [18]. Maybe, our approach can be combined with it.

• Parallel computing features.

• Implementations of hybrid clauses containing integer variables.

• Clause learning for integer variables.

• Can non-clause propagators report implications for their assignments?


Thank you for your attention.

[18] Olga Ohrimenko, Peter Stuckey, and Michael Codish. Propagation = Lazy Clause Generation. In Christian Bessiere, editor, Principles and Practice of Constraint Programming - CP 2007, 13th International Conference, CP 2007, Providence, RI, USA, September 23-27, 2007, Proceedings, volume 4741 of Lecture Notes in Computer Science. Springer, 2007.

[2] Krzysztof Apt. Principles of Constraint Programming. Cambridge University Press, 2003.

[9] Niklas Eén and Niklas Sörensson. An Extensible SAT-solver. In Enrico Giunchiglia and Armando Tacchella, editors, Theory and Applications of Satisfiability Testing - SAT 2003, 6th International Conference, Santa Margherita Ligure, Italy, May 5-8, 2003, volume 2919 of Lecture Notes in Computer Science, pages 502–518. Springer, 2004.

[3] Gilles Audemard, Lucas Bordeaux, Youssef Hamadi, Said Jabbour, and Lakhdar Sais. A Generalized Framework for Conflict Analysis. In Marques-Silva and Sakallah [16], pages 21–27.

[16] João Marques-Silva and Karem A. Sakallah, editors. Theory and Applications of Satisfiability Testing - SAT 2007, 10th International Conference, Lisbon, Portugal, May 28-31, 2007, volume 4501 of Lecture Notes in Computer Science. Springer, 2007.[17] Matthew W. Moskewicz, Conor F. Madigan, Ying Zhao, Lintao Zhang, and Sharad Malik. Chaff: Engineering an Efficient SAT Solver. In Proceedings of the 38th Design Automation Conference, DAC 2001, Las Vegas, NV, USA, June 18-22, 2001. ACM, 2001.

[20] Knot Pipatsrisawat and Adnan Darwiche. A Lightweight Component Caching Scheme for Satisfiability Solvers. In Marques-Silva and Sakallah [16], pages 294–299[25] The Gecode Team. Gecode: generic constraint development environment, 2008. Available at http://www.gecode.org.

References:
http://www.gecode.orghttp://www.gecode.org

Documents

Reconciling - Universität des Saarlandesraphael/bachelor/downloads/talk4...separator entry denotes branching level Depth-ﬁrst search explores a single path in a search tree, represented