Analysis of Branching Heuristics in SAT Solvers

This content delves into various branching heuristics used in SAT solvers, such as Exponential Recency Weighted Average, Conflict History-Based, and Tie-break of VSIDS. It discusses the decision-making processes of solvers and compares different approaches to handle ties and improve solver performance.

• Branching heuristics
• SAT solvers
• Tie-break
• VSIDS
• Conflict history-based

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1. Tie break University of Tokyo Seongsoo Moon

2. Lists of solvers Sequential solvers CHBR_GLUCOSE CHBR_GLUCOSE_TUNED TB_GLUCOSE TC_GLUCOSE (bug exist?) } Glucose + CHB [Liang+, 16] Glucose + TBVSIDS Glucose + CHB + TBVSIDS Parallel solver ParaGlueminisat Session 4A (12:00 12:30)

3. CHB [Liang+, AAAI 16] Conflict history-based branching heuristic Introduced in AAAI 2016 Title: Exponential Recency Weighted Average Branching Heuristic for SAT Solvers Solve significantly more instances than VSIDS

4. TBVSIDS [MOON+, CP-DP 2016] Tie break of VSIDS Tie occurs frequently and broken randomly in VSIDS Ties are broken randomly by default, although this is configurable [Moskewicz+, 2001] VSIDS in many SAT solvers uses heap array for VSIDS Proposal of meaningful selection from ties

5. VSIDS 2v Variables = { , , , , , } v v v v v v 1 2 3 4 5 6 3v 5v = ( ) 2 activity v 1 = ( ) 100 activity v 1v 4v 6v 2 = ( ) 19 activity v 3 = ( ) 25 activity v Pick 2 v reconstruc , t 4 = ( ) 36 activity v 5 = ( ) 3 activity v 5v 6 3v 4v 1v 6v

6. VSIDS 2v Variables = { , , , , , } v v v v v v 1 2 3 4 5 6 3v 5v = ( ) 2 activity v 1 = ( ) 100 activity v 1v 6v 4v 2 = ( ) 100 activity v 3 = ( ) 25 activity v Is this reasonable? Pick 2 v reconstruc , t 4 = ( ) 36 activity v 5 = ( ) 100 activity v 5v 6 3v 4v 1v 6v

7. TBVSIDS [MOON+, CP-DP 2016] 2v Variables = { , , , , , } v v v v v v 1 2 3 4 5 6 3v 5v = = ( ) , 2 ( ) 1 activity v activityQ v 1 1 = = ( ) 100 , ( ) 15 activity v activityQ v 1v 6v 4v 2 2 = = ( ) 100 , ( ) 10 activity v activityQ v 3 3 = = ( ) 25 , ( ) 10 activity v activityQ v VSIDS TBVSIDS 4 4 = = ( ) 36 , ( ) 5 activity v activityQ v 5 5 6v = = ( ) 100 , ( ) 20 activity v activityQ v 6 6 5v 2v Compare first by activity If activitys are equal, compare activityQ 1v 3v 4v

8. TBVSIDS [MOON+, CP-DP 2016] 2v Variables = { , , , , , } v v v v v v 1 2 3 4 5 6 3v 5v = = ( ) , 2 ( ) 1 activity v activityQ v 1 1 = = ( ) 100 , ( ) 15 activity v activityQ v 1v 6v 4v 2 2 = = ( ) 100 , ( ) 10 activity v activityQ v 3 3 = = ( ) 25 , ( ) 10 activity v activityQ v VSIDS TBVSIDS 4 4 = = ( ) 36 , ( ) 5 activity v activityQ v 5 5 6v = = ( ) 100 , ( ) 20 activity v activityQ v 6 6 Priority 5v 2v = = In VSIDS, In TBVSIDS, v v v 3 2 6 1v 3v 4v v v v 3 2 6

9. TBVSIDS [MOON+, CP-DP 2016] 2v Variables = { , , , , , } v v v v v v 1 2 3 4 5 6 3v 5v = = ( ) , 2 ( ) 1 activity v activityQ v 1 1 = = ( ) 100 , ( ) 20 activity v activityQ v 1v 6v 4v 2 2 = = ( ) 100 , ( ) 10 activity v activityQ v 3 3 = = ( ) 25 , ( ) 10 activity v activityQ v VSIDS TBVSIDS 4 4 = = ( ) 36 , ( ) 5 activity v activityQ v 5 5 6v = = ( ) 100 , ( ) 20 activity v activityQ v 6 6 Priority 5v 2v = = In VSIDS, In TBVSIDS, v v v 3 2 6 1v 3v 4v = v v v 3 2 6

10. TBVSIDS [MOON+, CP-DP 2016] Conflict v { , } 6v v { , } v 8 7 8 v v { , } { , , } v v v Update each varia ble`s activity 6 7 2 5 6 relevant t resolution o v { , , } { , , } v v v v v 2 5 7 3 5 7 { , , } v v v 2 3 5 Clause learning Update each varia ble`s activityQ { , , } v v v learned a in clause 2 3 5

11. TBVSIDS1 [MOON+, CP-DP 2016]

12. TBVSIDS2 [MOON+, CP-DP 2016]

13. Tie occurences Benchmarks: 300 instances from SAT Competition 2014 application track Timeout: 1,000 s 96 / 145 (66.2 %) reduced 110 / 151 (72.8 %) reduced

14. Hybrid CHB + TBVSIDS

15. Why Hybrid? Benchmarks: 300 instances from SAT Competition 2014 application track Timeout: 3,600 s

16. Results Benchmarks: 900 instances from 2014Crafted, 2014App and 2015App