Preprocessing in Pseudo-Boolean Optimization: An Experimental Evaluation

Transcription

1 Preprocessing in Pseudo-Boolean Optimization: An Experimental Evaluation Ruben Martins Inês Lynce Vasco Manquinho IST/INESC-ID, Technical University of Lisbon, Portugal 20/09/2009 Lisbon, Portugal

2 Motivation Preprocessing is a powerful tool in SAT solving: e.g. HyPre, NiVER, SatELite, ReVivAl, exthypre,... PBO is closely related to SAT However, preprocessing is almost non existent in PBO We propose to benefit from the advances in SAT preprocessing and use them in PBO In this talk, we will show how we can use HyPre and exthypre to preprocess a PBO instance

3 HyPre in PBO Initial PB Formula (1) (3) (4) Clausal Translate CNF Clauses HyPre Part to CNF formula Simplified CNF formula Cardinality + (2) PB Constraints (5) Simplify the PB Part If Eq. Variables (7) Translate to PB PB Part (6) Simplified Add Unit If fixed Clausal Clauses Variables Part (8) (9) Join Join

4 HyPre in PBO x1 V x2 V x3 ~x1 V x4 ~x2 V x4 ~x3 V x4

5 HyPre in PBO x1 V x2 V x3 x1 V x2 V x3 x1 V x2 V x3 x1 V x2 V x3 ~x3 V x4

6 Background (SAT and PBO) SAT (Boolean Satisfiability) CNF formula ϕ = x j SAT problem is to dedice if i ϕ i is satisfiable PBO (Pseudo-Boolean Optimization) PB constraints ϕ = j a jx j b Linear objective function f = j c jx j PBO problem is to find an assignment to x j such that all ϕ i are satisfied and the value of f is optimized

7 Background (PB Constraints) PB constraints can be divided into three categories: cardinality constraints: e.g. x 1 + x 2 + x 3 + x 4 2 clauses: e.g. x 1 + x 2 + x 3 1 general PB constraint: e.g. 7x 1 + 4x 2 + 5x 3 7 This categories form two distinct parts: The PB part is formed by the conjunction of cardinality constraints and general PB constraints The clausal part is formed by the conjunction of clauses

8 HyPre Preprocessor HyPre is based on Hyper-Binary Resolution (HypBinRes): (l 1... l n ) (l 1 y)... (l n 1 y) (y l n ) for n 2 In addition to HypBinRes, HyPre also simplifies the formula through Equality Reduction and Unit Propagation.

9 Initial PB Formula HyPre in PBO

10 HyPre in PBO Initial PB Formula (1) Clauses Clausal Part

11 HyPre in PBO Initial PB Formula (1) Clauses Clausal Part Cardinality + (2) PB Constraints PB Part

12 HyPre in PBO Initial PB Formula (1) (3) Clausal Translate Clauses Part to CNF CNF formula Cardinality + (2) PB Constraints PB Part

13 HyPre in PBO Initial PB Formula (1) (3) (4) Clausal Translate CNF Clauses HyPre Part to CNF formula Simplified CNF formula Cardinality + (2) PB Constraints PB Part

14 HyPre in PBO Initial PB Formula (1) (3) (4) Clausal Translate CNF Clauses HyPre Part to CNF formula Simplified CNF formula Cardinality + (2) PB Constraints (5) Simplify the PB Part If Eq. Variables PB Part

15 HyPre in PBO Initial PB Formula (1) (3) (4) Clausal Translate CNF Clauses HyPre Part to CNF formula Simplified CNF formula Cardinality + (2) PB Constraints PB Part (5) Simplify the PB Part (6) Add Unit Clauses If Eq. Variables If fixed Variables

16 HyPre in PBO Initial PB Formula (1) (3) (4) Clausal Translate CNF Clauses HyPre Part to CNF formula Simplified CNF formula Cardinality + (2) PB Constraints (5) Simplify the PB Part If Eq. Variables (7) Translate to PB PB Part (6) Simplified Add Unit If fixed Clausal Clauses Variables Part

17 HyPre in PBO Initial PB Formula (1) (3) (4) Clausal Translate CNF Clauses HyPre Part to CNF formula Simplified CNF formula Cardinality + (2) PB Constraints (5) Simplify the PB Part If Eq. Variables (7) Translate to PB PB Part (6) Simplified Add Unit If fixed Clausal Clauses Variables Part (8) (9) Join Join

18 exthypre Preprocessor exthypre is based on Extended Hyper Resolution (ExtHypRes), which is an extension of HypBinRes: (l 1 l 2... l n ) (l 1 α 1 ),..., (l n α n ) ( 1 i n (α i)) where α i for 1 i n are sub-clauses. exthypre produces new resolvent clauses that are added to the original formula

19 New graph representation of a CNF formula exthypre is applied through a new graph representation of a CNF formula: The idea behind this representation is to extend the binary implication graph for clauses of any size x1 {x3} x2 x1 V x2 V x3 x1 V x4 x4 x2 {x1} x3 x4 x2 V x4 x3 V x4 x3 {x2} x1

20 New graph representation of a CNF formula exthypre is applied through a new graph representation of a CNF formula: The idea behind this representation is to extend the binary implication graph for clauses of any size x1 {x3} x2 x1 V x2 V x3 x1 V x4 x4 x2 {x1} x3 x4 x2 V x4 x3 V x4 x3 {x2} x1

21 New graph representation of a CNF formula exthypre is applied through a new graph representation of a CNF formula: The idea behind this representation is to extend the binary implication graph for clauses of any size x1 {x3} x2 x1 V x2 V x3 x1 V x4 x4 x2 {x1} x3 x4 x2 V x4 x3 V x4 x3 {x2} x1

22 New graph representation of a CNF formula exthypre is applied through a new graph representation of a CNF formula: The idea behind this representation is to extend the binary implication graph for clauses of any size x1 {x3} x2 x1 V x2 V x3 x1 V x4 x4 x2 {x1} x3 x4 x2 V x4 x3 V x4 x3 {x2} x1

23 New graph representation of a CNF formula exthypre is applied through a new graph representation of a CNF formula: The idea behind this representation is to extend the binary implication graph for clauses of any size x1 {x3} x2 x1 V x2 V x3 x1 V x4 x4 x2 {x1} x3 x4 x2 V x4 x3 V x4 x3 {x2} x1

24 exthypre in PBO Enrich the graph representation: Information from cardinality constraints: Given x 1,..., x n b Suppose that n b + 1 literals are falsified Then if one of the b + 1 literals is falsified it implies that all the remaining b literals are satisfied x 1 + x 2 + x 3 + x 4 2 x 1 {x2} x 3 x 1 {x2} x 4 x 2 {x3} x 4 x 2 {x3} x 1 x 3 {x4} x 1 x 3 {x4} x 2 x 4 {x1} x 2 x 4 {x1} x 3

25 exthypre in PBO Enrich the graph representation: Information from general PB constraints: Probing on each literal of each general PB constraint 2x 1 + x 2 + x 3 2 x 1 x 2 x 1 x 3

26 Clause Selection Heuristics exthypre uses two fixed parameters: maximum size of the resolvent : 75 percentage of clauses to be considered: 10% Heuristics for clause selection: random - exthypre(75,10,r) larger clauses - exthypre(75,10,c) clauses with higher weight - exthypre(75,10,d)

27 Experimental Results (HyPre I) Average size of the 309 instances to be preprocessed by HyPre and average time of preprocessing Benchmark # Vars Constraints Cls Card PB Time aksoy 79 15, , logic-synthesis 74 1, , primes-dimacs-cnf 130 1, , routing , synthesis-ptl-cmos haplotype 8 19, ,186, ,

28 Experimental Results (HyPre II) Impact of HyPre preprocessing on bsolo, Pueblo and minisat+ Benchmark # bsolo Pueblo minisat+ w/o pre HyPre w/o pre HyPre w/o pre HyPre aksoy logic-synthesis primes-dimacs-cnf routing synthesis-ptl-cmos haplotype Total

29 Experimental Results (HyPre III) Details of some instances of HyPre preprocessing on bsolo Benchmark Instance bsolo w/o pre HyPre f20c10b 016 area delay fir07 area delay aksoy fir09 area opers fir08 trarea ac f20c10b 029 area delay fir07 area opers logic-synthesis prom2.pi routing s pb 64 64

30 Experimental Results (HyPre III) Details of some instances of HyPre preprocessing on Pueblo Benchmark Instance Pueblo w/o pre HyPre logic-synthesis sao2.b ssa primes-dimacs-cnf ssa ssa ssa routing s pb 64 64

31 Experimental Results (HyPre III) Details of some instances of HyPre preprocessing on minisat+ Benchmark primes-dimacs-cnf Instance minisat+ w/o pre HyPre ssa ssa ssa ssa

32 Experimental Results (exthypre I) Average time in seconds of exthypre preprocessing using the different clause selection heuristics Benchmark # exthypre exthypre exthypre (75,10,c) (75,10,d) (75,10,r) aksoy logic-synthesis primes-dimacs-cnf radar routing synthesis-ptl-cmos testset ttp

33 Experimental Results (exthypre II) Impact of exthypre preprocessing on bsolo Benchmark # w/o pre bsolo exthypre exthypre exthypre (75,10,c) (75,10,d) (75,10,r) aksoy logic-synthesis primes-dimacs-cnf radar routing synthesis-ptl-cmos testset ttp Total

34 Experimental Results (exthypre II) Details on some instances of exthypre preprocessing on bsolo: Benchmark aksoy primes-dimacs-cnf routing Instance w/o pre bsolo exthypre exthypre exthypre (75,10,c) (75,10,d) (75,10,r) fir07 area delay fir09 area opers fir09 area partials ii32b ii32e4.opb s pb s pb

35 Experimental Results (exthypre III) Impact of exthypre preprocessing and Pueblo Benchmark # w/o pre Pueblo exthypre exthypre exthypre (75,10,c) (75,10,d) (75,10,r) aksoy logic-synthesis primes-dimacs-cnf radar routing synthesis-ptl-cmos testset ttp Total

36 Experimental Results (exthypre III) Details on some instances of exthypre preprocessing on Pueblo: Benchmark logic-synthesis primes-dimacs-cnf Instance w/o pre Pueblo exthypre exthypre exthypre (75,10,c) (75,10,d) (75,10,r) C880.a sao2.b hanoi ii32c ii32e

37 Experimental Results (exthypre IV) Impact of exthypre preprocessing on minisat+ Benchmark # w/o pre minisat+ exthypre exthypre exthypre (75,10,c) (75,10,d) (75,10,r) aksoy logic-synthesis primes-dimacs-cnf radar routing synthesis-ptl-cmos testset ttp Total

38 Experimental Results (exthypre IV) Details on some instances of exthypre preprocessing on minisat+: Benchmark Instance w/o pre minisat+ exthypre exthypre exthypre (75,10,c) (75,10,d) (75,10,r) fir05 area delay aksoy matrix 5x matrix 5x logic-synthesis 5xp1.b primes-dimacs-cnf ii32d

39 Conclusions & Future Work Applying SAT techniques in PBO preprocessing lead to a few improvements This take us to believe that further research can lead to better results In the future we plan to improve the efficiency of our PBO version of exthypre and study different heuristics We also plan to incorporate other SAT techniques into PBO preprocessing