improving maximum satisfiability solving with preprocessing

Improving Maximum Satisfiability solving withpreprocessing

Data61 Decision Sciences SeminarJeremias Berg

University of Helsinki

October 26, 2018

J. Berg (University of Helsinki) MaxSAT preprocessing October 26, 2018 1 / 24

Who am I?

29-year old postdoc from Finland.Here to meet new people and work onnew problems.

Graduated with distinction from theUniversity of Helsinki in June 2018.

Supervisor: Prof Matti Jarvisalo.

MSc, in mathematics 2014In Melbourne until Christmas, bestchance to find me in Monash on aThursday.www.jeremiasberg.com

Take Home Messages

Preprocessing, a currently underused approach to MaxSATsolving.In contrast to SAT solving, a naive integration of preprocessing inMaxSAT can hurt solvers.Smarter integration changes conceptualisation of MaxSAT.Talk is mainly based on: Jeremias Berg, Paul Saikko, and MattiJarvisalo. 2015. Improving the effectiveness of SAT-basedpreprocessing for MaxSAT. In Proc IJCAI’15, AAAI Press 239-245.

Outline

Combinatorial optimization problemsMaximum SatisfiabilitySolving Combinatorial optimization problems with MaxSAT.Preprocessing in SAT and MaxSAT solvingIntegrating MaxSAT preprocessing and solving”New” conceptualisation of MaxSAT

Combinatorial Optimization

Find a solution that minimizes a given cost function from a finite setof feasible solutions.

Encountered in many domains, including:Route PlanningPackingSchedulingMachine Learning / Data Analysis / Artificial Intelligence:

itemset mining, clustering, classification, probabilistic modelling,data visualization etc.

NP-hardnessMost interesting problems are difficult to solve!

Combinatorial Optimization

Find a solution that minimizes a given cost function from a finite setof feasible solutions.

Encountered in many domains, including:Route PlanningPackingSchedulingMachine Learning / Data Analysis / Artificial Intelligence:

itemset mining, clustering, classification, probabilistic modelling,data visualization etc.

NP-hardnessMost interesting problems are difficult to solve!

Solving Difficult Combinatorial Optimization Problems

Roughly four different approachesApproximation algorithms:Local search algorithms:Problem specific exact solution algorithms:Exact declarative methods:

Roughly four different approachesApproximation algorithms:

scalable (fast) algorithms,not guaranteed to return the optimal solution,mathematically provable guarantees on the quality of the obtainedsolution.

Local search algorithms:Problem specific exact solution algorithms:Exact declarative methods:

Roughly four different approachesApproximation algorithms:Local search algorithms:

very scalable algorithms,traverse the search space moving from solution to solution,no guarantees on the quality of solution obtained.

Problem specific exact solution algorithms:Exact declarative methods:

Roughly four different approachesApproximation algorithms:Local search algorithms:Problem specific exact solution algorithms:

less scalable,guaranteed to find the optimal solution,designed for the specific problem at hand.

Exact declarative methods:

Roughly four different approachesApproximation algorithms:Local search algorithms:Problem specific exact solution algorithms:Exact declarative methods:

model the problem instance with mathematical constraints,solve the constraint instance instead.in this talk: Maximum Satisfiability

(Weighted Partial) Maximum Satisfiability (MaxSAT)

Boolean variables and negations (literals):

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x =y =Truez =Falset =Boolean Satisfiability (SAT):

Decide the existence of an assign-ment that satisfies all clauses.

In this case UNSAT

(Weighted Partial) Maximum Satis-fiability (MaxSAT):Find an assignment that satisfies allhard clauses and a maximum sum ofweights of soft clauses.

Cost of solution: 5

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x =True

In this case UNSAT

Cost of solution: 5

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x =False

In this case UNSAT

Cost of solution: 5

Clause:

x ∨ ¬y ∨ z ∨ ¬t

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

In this case UNSAT

Cost of solution: 5

Clause:

Satisfied

x ∨ ¬y ∨ z ∨ ¬t

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x =Truey =Truez =Falset =False

Boolean Satisfiability (SAT):Decide the existence of an assign-ment that satisfies all clauses.

In this case UNSAT

Cost of solution: 5

Clause:

Falsified

x ∨ ¬y ∨ z ∨ ¬t

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x =Falsey =Truez =Falset =True

In this case UNSAT

Cost of solution: 5

CNF-Formula:

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

In this case UNSAT

Cost of solution: 5

CNF-Formula:

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x =y =Truez =Falset =

In this case UNSAT

Cost of solution: 5

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

In this case UNSAT

Cost of solution: 5

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2

¬x1 ∨ ¬y1

¬x1 ∨ ¬z1

¬y1 ∨ ¬z1

¬x2 ∨ ¬y2

¬x2 ∨ ¬z2

¬y2 ∨ ¬z2

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

x1 ∨ x2

y1 ∨ y2

z1 ∨ z2(¬x1 ∨ ¬y1,3)(¬x1 ∨ ¬z1,2)(¬y1 ∨ ¬z1,7)(¬x2 ∨ ¬y2,1)(¬x2 ∨ ¬z2,2)(¬y2 ∨ ¬z2,5)

In this case UNSAT

Cost of solution: 5

Combinatorial Optimization with MaxSAT

Instance p of P MaxSAT instance F(p)

Solution τ to F(p)Solution to p

MaxSAT encoding

MaxSAT solver

Reconstruction

Why MaxSAT?

Combinatorial Optimization with MaxSAT andpreprocessing

Instance p of P MaxSAT instance F(p)

Preprocessed MaxSAT instance pre(F(p))

Solution τp to pre(F(p))

Solution to p

Solution τ to F(p)

MaxSAT encoding

MaxSAT preprocessing

MaxSAT solverMaxSATReconstruction

Reconstruction

Preprocessing in SAT solving

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(¬a ∨ b)

(¬d ∨ ¬e)

((((((((a ∨ ¬b ∨ ¬a)

(¬a ∨ b)

(¬a ∨ d)(¬a ∨ e)

(¬d ∨ ¬e)

(¬a ∨ b)(¬a ∨ d)(¬a ∨ e)(¬d ∨ ¬e) ()

Simple: Removal of subsumed clausesMore involved: Elimination of variables by resolvingFrom now on: SAT-based preprocessing rules

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

((((((((c ∨ ¬d ∨ ¬e)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(¬a ∨ b)

(¬d ∨ ¬e)

((((((((a ∨ ¬b ∨ ¬a)

(¬a ∨ b)

(¬a ∨ d)(¬a ∨ e)

(¬d ∨ ¬e)

(¬a ∨ b)(¬a ∨ d)(¬a ∨ e)(¬d ∨ ¬e) ()

Simple: Removal of subsumed clauses

More involved: Elimination of variables by resolvingFrom now on: SAT-based preprocessing rules

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

((((((((c ∨ ¬d ∨ ¬e)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

((((((((a ∨ ¬b ∨ ¬a)

(¬a ∨ b)

(¬a ∨ d)(¬a ∨ e)

(¬d ∨ ¬e)

(¬a ∨ b)(¬a ∨ d)(¬a ∨ e)(¬d ∨ ¬e) ()

Simple: Removal of subsumed clauses

More involved: Elimination of variables by resolvingFrom now on: SAT-based preprocessing rules

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

((((((((c ∨ ¬d ∨ ¬e)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

((((((((a ∨ ¬b ∨ ¬a)

(¬a ∨ b)

(¬a ∨ d)(¬a ∨ e)

(¬d ∨ ¬e)

(¬a ∨ b)(¬a ∨ d)(¬a ∨ e)(¬d ∨ ¬e) ()

Simple: Removal of subsumed clausesMore involved: Elimination of variables by resolving

From now on: SAT-based preprocessing rules

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

((((((((c ∨ ¬d ∨ ¬e)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(¬a ∨ b)

(¬d ∨ ¬e)

((((((((a ∨ ¬b ∨ ¬a)

(¬a ∨ b)

(¬a ∨ d)(¬a ∨ e)

(¬d ∨ ¬e)

(a)(¬a ∨ b)(¬a ∨ d)(¬a ∨ e)(¬d ∨ ¬e)

Simple: Removal of subsumed clausesMore involved: Elimination of variables by resolving

From now on: SAT-based preprocessing rules

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

((((((((c ∨ ¬d ∨ ¬e)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(a)(a ∨ ¬b ∨ ¬c)

(¬a ∨ b)(¬a ∨ c)

(¬c ∨ d)(¬c ∨ e)(¬d ∨ ¬e)

(¬a ∨ b)

(¬d ∨ ¬e)

((((((((a ∨ ¬b ∨ ¬a)

(¬a ∨ b)

(¬a ∨ d)(¬a ∨ e)

(¬d ∨ ¬e)

(a)(¬a ∨ b)(¬a ∨ d)(¬a ∨ e)(¬d ∨ ¬e) ()

Simple: Removal of subsumed clausesMore involved: Elimination of variables by resolvingFrom now on: SAT-based preprocessing rules

SAT-based preprocessing in MaxSAT [Belov et al., 2013]

(x ∨ y)(¬y ∨ z)

(x ∨ z)SAT

(x ∨ ¬y)(x ∨ y)

(¬x)(x)

(¬x)(x)M

AT Har

Directly applying SAT-based preprocessing rules does not work.Solution: Switch ”softness” from clauses to literals.

Satisfiability is Preserved

Solutions are not

SAT-Based preprocessing in MaxSAT

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)

((a),3)((¬d ∨ ¬e),5)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(s1 ∨ s2)(s1 ∨ s2)

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

1) Add soft literals to all soft clauses2) Run (restricted) SAT-based preprocessing on all clauses3) Add unit soft clauses for each soft literal

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)

((a),3)((¬d ∨ ¬e),5)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(s1 ∨ s2)(s1 ∨ s2)

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

1) Add soft literals to all soft clauses

2) Run (restricted) SAT-based preprocessing on all clauses3) Add unit soft clauses for each soft literal

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)

((a),3)((¬d ∨ ¬e),5)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(s1 ∨ s2)(s1 ∨ s2)

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

1) Add soft literals to all soft clauses

2) Run (restricted) SAT-based preprocessing on all clauses3) Add unit soft clauses for each soft literal

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)

((a),3)((¬d ∨ ¬e),5)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

1) Add soft literals to all soft clauses2) Run (restricted) SAT-based preprocessing on all clauses

3) Add unit soft clauses for each soft literal

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)

((a),3)((¬d ∨ ¬e),5)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

1) Add soft literals to all soft clauses2) Run (restricted) SAT-based preprocessing on all clauses

3) Add unit soft clauses for each soft literal

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)

((a),3)((¬d ∨ ¬e),5)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(a ∨ ¬b ∨ ¬c)(¬a ∨ b)(¬a ∨ c)

(c ∨ ¬d ∨ ¬e)(¬c ∨ d)(¬c ∨ e)(a ∨ s1)

(¬d ∨ ¬e ∨ s2)

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

1) Add soft literals to all soft clauses2) Run (restricted) SAT-based preprocessing on all clauses3) Add unit soft clauses for each soft literal

Effect of preprocessing

However...[Berg et al., 2015a]

Hypothesis for poor performance[Berg et al., 2015a]

SAT solving: Ineffective preprocessing does not hurt solver.MaxSAT solving: Ineffective preprocessing adds (unnecessary)extra clauses and variables.

Variables ClausesInstance Original Preprocess Original Preprocessmaxcut.wcnf 41 1681 = (41 + 1640) 1640 3280 = (2× 1640)

Hypothesis for poor performance[Berg et al., 2015a]

SAT solving: Ineffective preprocessing does not hurt solver.MaxSAT solving: Ineffective preprocessing adds (unnecessary)extra clauses and variables.

Variables ClausesInstance Original Preprocess Original Preprocessmaxcut.wcnf 41 1681 = (41 + 1640) 1640 3280 = (2× 1640)

Detour: SAT-based MaxSAT solving

Reduce MaxSAT solving to a sequence of satisfiability computations.

Linear Search: Iteratively strengthen anupper bound (UB) on the known optimal cost.

Core Guided: Iteratively strengthen anlower bound by extracting usatisfiable subsets (cores)of soft clauses.

Implicit hitting set: Iteratively extract coresand compute minimum costhitting sets over them.

Take Home Message:Most SAT-based solvers addassumption variablesto soft clauses

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

CNF(3a1 + 5a2 ≤ UB)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

κ = {a1,a2}Increase LB by 3

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

κ = {a1,a2}MIN-HS = {a1}

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

CNF(3a1 + 5a2 ≤ UB)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

κ = {a1,a2}MIN-HS = {a1}

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

CNF(3a1 + 5a2 ≤ UB)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

κ = {a1,a2}MIN-HS = {a1}

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

CNF(3a1 + 5a2 ≤ UB)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

κ = {a1,a2}MIN-HS = {a1}

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

CNF(3a1 + 5a2 ≤ UB)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

κ = {a1,a2}MIN-HS = {a1}

Integrating MaxSAT preprocessing and solving[Berg et al., 2015a], IJCAI

Summary:Preprocessing adds:

A soft literal to each original soft clauseAn (explicit) soft clause for each soft literal

MaxSAT solver adds:An assumption variable to each soft clauseAn (implicit) soft clause for each assumption variable

Idea:Reuse soft literals as assumption variables.Soft clauses of the preprocessed instance not needed at all.

Integrating MaxSAT preprocessing and solving[Berg et al., 2015a], IJCAI

Summary:Preprocessing adds:

A soft literal to each original soft clauseAn (explicit) soft clause for each soft literal

MaxSAT solver adds:An assumption variable to each soft clauseAn (implicit) soft clause for each assumption variable

Idea:Reuse soft literals as assumption variables.Soft clauses of the preprocessed instance not needed at all.

MaxSAT solving Integrated with preprocessing

(s1 ∨ s2)

((¬s1),3)((¬s2),5)

(s1 ∨ s2)

��

�(¬s1 ∨ a1)

��(¬s2 ∨ a2)

(((((((

CNF(3a1 + 5a2 ≤ UB)

CNF(3s1 + 5s2 ≤ UB)

Results

Conceptual change to MaxSAT

Originally:one-to-one mapping be-tween assumption variablesand soft clauses

When reusing:Can have clauses withmany soft literalsSoft literals can be in manyclauses.

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

CNF(3a1 + 5a2 ≤ UB)

��(¬s1 ∨ a1)

��(¬s2 ∨ a2)

((((((((

((((CNF(3a1 + 5a2 ≤ UB)

CNF(3s1 + 5s2 ≤ UB)

Conceptual change to MaxSAT

Originally:one-to-one mapping be-tween assumption variablesand soft clauses

When reusing:Can have clauses withmany soft literalsSoft literals can be in manyclauses.

(s1 ∨ s2)

(¬s1 ∨ a1)

(¬s2 ∨ a2)

CNF(3a1 + 5a2 ≤ UB)

��(¬s1 ∨ a1)

��(¬s2 ∨ a2)

((((((((

((((CNF(3a1 + 5a2 ≤ UB)

CNF(3s1 + 5s2 ≤ UB)

Implications of the idea

”Better” definition of MaxSATINPUT: CNF-formula F , weight for all variables x of FOBJECTIVE: Find τ satisfying F and minimizing

∑x∈VAR(F ) w(x)τ(x).

Building on the definition:Identify reusable literals from the input instanceitself [Berg et al., 2015b].Decrease number of soft literals [Korhonen et al., 2017].MaxSAT specific preprocessing techniques [Berg et al., 2016]

0 10 20 30 40 50 60 70 80 90 100

Soft C

lauses R

eused (

Instances (%)

IndustrialCrafted

C1 ∨ s1C2 ∨ s2

C1 ∨ s1C2 ∨ s1

If C1 ∨ C2 tautology andw(s1) = w(s2)

Fixes earlier problem instance:

Variables ClausesInstance Original Preprocess Preprocess+ Original Preprocess Preprocess+maxcut.wcnf 41 1681 861 1640 3280 2460

s1 ∨ s2 s1

If w(s1) ≤ w(s2)

Conclusions

MaxSAT for solving NP-hard optimization problemsPreprocessing underused in MaxSAT solving.Integrating preprocessing is difficult when optimizing clauses

Solution: optimize literals instead.Interesting insights to MaxSAT

Thank you!

Try MaxPre, our preprocessor at:https://www.cs.helsinki.fi/group/coreo/maxpre/

References I

Belov, A., Morgado, A., and Marques-Silva, J. (2013).SAT-based preprocessing for MaxSAT.In Proceedings of the 19th International Conference on Logic forProgramming, Artificial Intelligence, and Reasoning, volume 8312of Lecture Notes in Computer Science, pages 96–111. Springer.

Berg, J., Saikko, P., and Jarvisalo, M. (2015a).Improving the effectiveness of SAT-based preprocessing forMaxSAT.In Proceedings of the 24th International Joint Conference onArtificial Intelligence, pages 239–245. AAAI Press.

References II

Berg, J., Saikko, P., and Jarvisalo, M. (2015b).Re-using auxiliary variables for maxsat preprocessing.In 27th IEEE International Conference on Tools with ArtificialIntelligence, ICTAI 2015, Vietri sul Mare, Italy, November 9-11,2015, pages 813–820.

Berg, J., Saikko, P., Jarvisalo, M., et al. (2016).Subsumed label elimination for maximum satisfiability.In ECAI 2016 22nd European Conference on Artificial Intelligence,29 August–2 September 2016, The Hague, The Netherlands,Including Prestigious Applications of Artificial Intelligence (PAIS2016), Proceedings.

References III

Korhonen, T., Berg, J., Saikko, P., and Jarvisalo, M. (2017).MaxPre: An extended MaxSAT preprocessor.In Proceedings of the 20th International Conference on Theoryand Applications of Satisfiability Testing, volume 10491 of LectureNotes in Computer Science, pages 449–456. Springer.

improving maximum satisfiability solving with preprocessing

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