Improving Unsatisfiability-based Algorithms for Boolean Optimization

Transcription

1 Improving Unsatisfiability-based Algorithms for Boolean Optimization Vasco Manquinho Ruben Martins Inês Lynce IST/INESC-ID, Technical University of Lisbon, Portugal SAT 2010, Edinburgh 1 / 27

2 Motivation Increasing interest in generalizations of SAT SAT techniques extended for MaxSAT, PBO and WBO Unsatisfiability-based algorithms have been proposed for Boolean Optimization problems very effective for several classes of instances can perform poorly on instances that are easy for classical approaches Integration of procedures in a unique Boolean optimization framework 2 / 27

3 Outline Background MaxSAT, PBO and WBO Algorithmic Solutions Classical Approaches Unsatisfiability-based approaches Improving Unsatisfiability-based algorithms PBO as preprocessing Constraint Branching Experimental Results Conclusions 3 / 27

4 Maximum Satisfiability (MaxSAT) MaxSAT Problem Given a CNF formula ϕ, find an assignment to problem variables that maximizes the number of satisfied clauses in ϕ (or minimizes the number of unsatisfied clauses). Partial MaxSAT Problem Given a conjunction of two CNF formulas ϕ h and ϕ s, find an assignment to problem variables that satisfies all hard clauses (ϕ h ) and maximizes the number of satisfied soft clauses (ϕ s ). 4 / 27

5 Maximum Satisfiability (MaxSAT) Weighted CNF Formula set of weighted clauses weighted clause: pair (ω, c) where ω is a clause and c N is a positive cost of unsatisfying ω Weighted MaxSAT Problem Given a weighted CNF formula ϕ s,c, find an assignment to problem variables that minimizes the total cost of unsatisfied clauses. Weighted Partial MaxSAT Problem Given a weighted CNF formula ϕ s,c and a classical CNF formula ϕ h, find an assignment to problem variables that satisfies all hard clauses (ϕ h ) and minimizes the total cost of unsatisfied soft clauses in ϕ s,c. 5 / 27

6 Pseudo-Boolean Optimization (PBO) Pseudo-Boolean Optimization n minimize c j x j subject to j=1 n a ij l j b i, j=1 l j {x j, x j }, x j {0, 1}, a ij, b i, c j N / 27

7 Weighted Boolean Optimization (WBO) WBO Formula Weighted Boolean Optimization formula is composed of two pseudo-boolean constraint sets (ϕ h, ϕ s ): ϕ h : set of hard pseudo-boolean constraints ϕ s : set of soft weighted pseudo-boolean constraints Soft pseudo-boolean constraint (ω, c): ω: pseudo-boolean constraint there is an integer weight c representing the cost of not satisfying ω WBO Problem Given a WBO formula, find an assignment to problem variables that satisfies all hard constraints (ϕ h ) and minimizes the total cost of unsatisfied soft constraints (ϕ s ). 7 / 27

8 WBO (Example) Weighted Boolean Optimization instance ϕ h = {x 1 + x 2 + x 3 2, 2x 1 + x 2 + x 3 2} ϕ s = {(x 1 + x 2 1, 2), (x 1 + x 3 1, 3)} Assignments that satisfy all hard constraints: (1) x 1 = x 3 = 1 ; x 2 = 0 ; P c i = 3 (2) x 1 = 0 ; x 2 = x 3 = 1 ; P c i = 2 (solution) 8 / 27

9 Encode MaxSAT as WBO For each hard clause (l 1 l 2 l k ) define a hard PB constraint as l 1 + l l k 1 For each weighted soft clause (ω, c) where ω = (l 1 l 2 l k ) define a soft PB constraint as l 1 + l l k 1 with weight c 9 / 27

10 Encode MaxSAT as WBO (Example) Weighted Partial MaxSAT instance ϕ h = {x 1 x 2 x 3, x 2 x 3, x 1 x 3 } ϕ s = {(x 3, 5), (x 1 x 2, 3), (x 1 x 3, 2)} Corresponding WBO instance ϕ h = {x 1 + x 2 + x 3 1, x 2 + x 3 1, x 1 + x 3 1} ϕ s = {(x 3 1, 5), (x 1 + x 2 1, 3), (x 1 + x 3 1, 2)} 10 / 27

11 Encode PBO as WBO For each pseudo-boolean constraint n a ij l j b i add this PB constraint to the set of hard PB constraints For each term c j x j in the objective function add a weighted soft PB constraint of the form ((x j 1), c j ) j=1 11 / 27

12 Encode PBO as WBO (Example) Pseudo-Boolean Optimization instance Corresponding WBO instance minimize 4x 1 + 2x 2 + x 3 subject to 2x 1 + 3x 2 + 5x 3 5 x 1 + x 2 1 x 1 + x 2 + x 3 2 ϕ h = {2x 1 + 3x 2 + 5x 3 5, x 1 + x 2 1, x 1 + x 2 + x 3 2} ϕ s = {(x 1 1, 4), (x 2 1, 2), (x 3 1, 1)} 12 / 27

13 Algorithmic Solutions (Classical Approaches) Branch and bound: e.g. MaxSatz, MiniMaxSAT Iteration of the upper bound: e.g. Pueblo, minisat+ Conversions from one Boolean formalism to another: e.g. minisat+, SAT4J MS 13 / 27

14 Unsatisfiability-based MaxSAT Original algorithm proposed by Fu&Malik [SAT 2006]: (1) Identify unsatisfiable sub-formula of an UNSAT formula SAT solver able to generate an UNSAT core (2) For each unsatisfiable sub-formula ϕ C : Relax all soft clauses in ϕ C by adding a new relaxation variable to each clause Add a new constraint such that at most 1 relaxation variable is assigned value 1 (3) When the resulting CNF formula is SAT, the solver terminates (4) Otherwise, go back to 1 14 / 27

15 Unsatisfiability-based MaxSAT 1 ϕ W ϕ 2 while (ϕ W is UNSAT) 3 do Let ϕ C be an unsatisfiable sub-formula of ϕ W 4 V R 5 for each soft clause ω ϕ C 6 do ω R ω {r} 7 ϕ W ϕ W {ω} {ω R } 8 V R V R {r} 9 ϕ R CNF( r V R r = 1) Equals1 constraint 10 ϕ W ϕ W ϕ R Clauses in ϕ R are declared hard 11 return ϕ number of relaxation variables assigned to 1 15 / 27

16 Unsatisfiability-based Weighted MaxSAT 1 ϕ W ϕ 2 cost lb 0 3 while (ϕ W is UNSAT) 4 do Let ϕ C be an unsatisfiable sub-formula of ϕ W 5 min c min ω ϕc hard(ω) cost(ω) 6 cost lb cost lb + min c 7 V R 8 for each soft clause ω ϕ C 9 do ω R ω {r} 10 cost(ω R ) min c 11 if cost(ω) > min c 12 then ϕ W ϕ W {ω R } 13 cost(ω) cost(ω) min c 14 else ϕ W ϕ W {ω} {ω R } 15 V R V R {r} 16 ϕ W ϕ W CNF( r V R r = 1) 17 return cost lb 16 / 27

17 Unsatisfiability-based Weighted MaxSAT Weighted MaxSAT instance ϕ h = {x 1 x 2 x 3, x 2 x 3, x 1 x 3 } ϕ s = {(x 3, 5), (x 1 x 2, 3), (x 1 x 3, 2)} Unsatisfiable sub-formula: min C = 3 ϕ C = { x 2 x 3, x 1 x 3, (x 3, 5), (x 1 x 2, 3)} Relax (x 1 x 2, 3) to (r 1 x 1 x 2, 3) Split (x 3, 5) into (x 3, 2) and (r 2 x 3, 3) Add CNF(r 1 + r 2 = 1) to ϕ h 17 / 27

18 Unsatisfiability-based Weighted MaxSAT Weighted MaxSAT instance ϕ h = {x 1 x 2 x 3, x 2 x 3, x 1 x 3 } ϕ s = {(x 3, 5), (x 1 x 2, 3), (x 1 x 3, 2)} Results in a new formula: ϕ h = {x 1 x 2 x 3, x 2 x 3, x 1 x 3, CNF(r 1 + r 2 = 1)} ϕ s = {(x 3, 2), (r 2 x 3, 3), (r 1 x 1 x 2, 3), (x 1 x 3, 2)} 17 / 27

19 Algorithm for Weighted Boolean Optimization Follows the same approach as Unsatisfiability-Based Weighted MaxSAT algorithm Instead of SAT solver, uses Pseudo-Boolean solver enhanced with unsatisfiable sub-formula extraction Relaxation of pseudo-boolean constraints a j l j b b r + P a j l j b No need to encode constraint r V R r = 1 into CNF 18 / 27

20 Improving Unsatisfiability-based Algorithms Unsatisfiability-based algorithms search on the lower bound. Sometimes is better to search on the upper bound: (1) PBO as Preprocessing The number of relaxation variables grows significantly at each step: (2) Constraint Branching 19 / 27

21 Encode WBO as PBO For each hard PB constraint n a ij l j b i j=1 add this PB constraint to the set of constraints n For each weighted soft PB constraint a ij l j b i with cost c j j=1 define a PB constraint with a new relaxation variable r P b i r + n a ij l j b i j=1 add c j r to the objective function 20 / 27

22 Encode WBO as PBO (Example) Weighted Boolean Optimization instance ϕ h = {x 1 + x 2 + x 3 2, 2x 1 + x 2 + x 3 2, x 1 + x 4 1} ϕ s = {(x 1 + x 2 1, 2), (x 1 + x 3 1, 3), (x 4 1, 4)} Corresponding PBO instance minimize 2r 1 + 3r 2 + 4r 3 subject to x 1 + x 2 + x 3 2 2x 1 + x 2 + x 3 2 x 1 + x 4 1 r 1 + x 1 + x 2 1 r 2 + x 1 + x 3 1 r 3 + x / 27

23 PBO as Preprocessing (1) Simplification techniques are used in the PBO formula: a generalization of Hypre for PB formulas is used (2) The PBO formula is solved using tight limits: PB solver is used for 10% of the time limit If optimality is not proved, the formula is translated back to WBO Small learnt clauses are kept in the WBO formula as hard clauses 22 / 27

24 Using Constraint Branching Consider the following Equals1 constraint: k r i = 1: i=1 If r i is assigned to 1, all other variables r j r i must be 0 However, if r i is assigned to 0, no propagation occurs Assigning value 1 to any of these variables produces very different search trees 23 / 27

25 Using Constraint Branching Constraint Branching: Instead of assigning one variables, half of the variables are assigned: k/2 P ω c1 : r i = 0 i=1 If ϕ {ω c1} is unsatisfiable then: i r i = 1, with 1 i k 2 kp we can infer ω c2 : r i = 0 i=k/ / 27

26 Computing Cores with Constraint Branching compute core(ϕ) 1 if (no large Equals1 constraint exist in ϕ) 2 then (st, ϕ C ) PB(ϕ) 3 return (st, ϕ C ) 4 else Select a large Equals1 constraint ω from ϕ 5 k = size(ω) 6 ω c1 : k/2 i=1 r i = 0 7 (st, ϕ C1 ) compute core(ϕ {ω c1 }) 8 if (st = SAT ω c1 / ϕ C1 ) 9 then return (st, ϕ C1 ) 10 else ω c2 : k i=k/2+1 r i = 0 11 (st, ϕ C2 ) compute core(ϕ {ω c2 }) 12 if (st = SAT ω c2 / ϕ C2 ) 13 then return (st, ϕ C2 ) 14 else return (st, ϕ C1 ϕ C2 ) 24 / 27

27 Experimental Results Industrial benchmark sets of the partial MaxSAT problem The most effective MaxSAT solvers from the MaxSAT evaluation of 2009 were considered: MSUncore, SAT4J (MS), pm2 Timeout: 1800 seconds Intel Xeon 5160 server with 3GB RAM 25 / 27

28 Experimental Results Solved Instances for Industrial Partial MaxSAT: Benchmark set #I MSUncore SAT4J (MS) pm2 wbo1.0 wbo1.2 bcp-fir bcp-hipp-yra bcp-msp bcp-mtg bcp-syn CircuitTraceCompaction HaplotypeAssembly pbo-mqc pbo-routing PROTEIN INS Total / 27

29 Conclusions PBO solvers can be used as a preprocessing step such that: 1) inference preprocessing techniques are used; 2a) some problems are easily solved with a search on the upper bound; 2b) restrict the search space by learning hard constraints. Constraint branching can improve the effectiveness of the solver Experimental results show that these techniques significantly improve the performance of wbo These results provide a strong stimulus for further integration of other Boolean optimization techniques 27 / 27