Solving Optimization Problems via Maximum Satisfiability ... · Lectio Praecursoria Jeremias Berg J. Berg Lectio praecursoria 1 / 22. Thesis research Contributions to exact declarative

Solving Optimization Problems viaMaximum Satisfiability:

Encodings and Re-Encodings

Lectio PraecursoriaJeremias Berg

J. Berg Lectio praecursoria 1 / 22

Thesis research

Contributions to exact declarative solution methods to NP-hardcombinatorial optimization problems

More specificallySolver independent preprocessing for Maximum Satisfiability(MaxSAT)New MaxSAT encodings of NP-hard data analysis problems

Correlation ClusteringBounded Treewidth Bayesian Network Structure Learning


Thesis research

Contributions to exact declarative solution methods to NP-hardcombinatorial optimization problems

More specificallySolver independent preprocessing for Maximum Satisfiability(MaxSAT)New MaxSAT encodings of NP-hard data analysis problems

Correlation ClusteringBounded Treewidth Bayesian Network Structure Learning


Combinatorial Optimization

Find a solution from a finite set of feasible solutions that minimizesa given cost function.

4

32

17

86

3

4

17

8

3

32

6

2

7

6

3

1

4

3

8

The Travelling Salesperson Problem:

Instance:Set of locations & pairwise distancesFeasible Solutions:Routes that visit all locationsCost of Solutions:Length of route

Cost:




4

32

17

86

3

4

17

8

3

32

6

2

7

6

3

1

4

3

8



Cost:




4

32

17

86

3

4

17

8

3

32

6

2

7

6

3

1

4

3

8


Instance:Set of locations & pairwise distances

Feasible Solutions:Routes that visit all locationsCost of Solutions:Length of route

Cost:




1

25

63

4

4

32

17

86

3

4

17

8

3

32

6

2

7

6

3

1

4

3

8


Instance:Set of locations & pairwise distancesFeasible Solutions:Routes that visit all locations

Cost of Solutions:Length of route

Cost:




1

25

63

4

4

32

17

86

3

4

17

8

3

32

6

2

7

6

3

1

4

3

8



Cost: 23




1

62

35

4

4

32

17

86

3

4

17

8

3

32

6

2

7

6

3

1

4

3

8



Cost: 19


Combinatorial Optimization Problems are common

Route Plannning:Find the shortest route that visits all of a given set of locations

Packing:Pack wares in a manner that wastes the least amount of space

Scheduling:Design timetables that waste the least amount of time

Machine Learning / Data Analysis / Artificial Intelligence:itemset mining, clustering, classification, probabilistic modelling,data visualization etc.

NP-hardnessMost interesting problems are difficult to solve!


Combinatorial Optimization Problems are common

Route Plannning:Find the shortest route that visits all of a given set of locations

Packing:Pack wares in a manner that wastes the least amount of space

Scheduling:Design timetables that waste the least amount of time

Machine 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) algorithmsnot guaranteed to return the optimal solutionmathematically 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 algorithmstraverse the search space moving from solution to solutionno 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 scalableguaranteed to find the optimal solutiondesigned 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 constraintssolve the constraint instance instead


Constraint optimizationPropositional Logic

x

Boolean variable:

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =

x =y =Truez =Falset =

Maximum Satisfiability (MaxSAT):Find an assignment that satisfies allhard clauses and a maximum sum ofweights of soft clauses.

Cost of solution:



x

Boolean variable:

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =True



Cost of solution:



x

Boolean variable:

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =False



Cost of solution:



x ¬x

Literal:

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =



Cost of solution:



x ¬x

Literal:

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =True



Cost of solution:



x ¬x

Literal:

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =False



Cost of solution:



x

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =



Cost of solution:



x

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =

x =Truey =Truez =Falset =False


Cost of solution:



x

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =

x =Falsey =Truez =Falset =True


Cost of solution:



x

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =



Cost of solution:


Constraint optimizationMaximum Satisfiability (MaxSAT)

x

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =



Cost of solution:


Constraint optimizationMaximum Satisfiability (MaxSAT)

x

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)

Har

dS

oft

x1 ∨ x2

y1 ∨ y2

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

Har

dS

oft

x =



Cost of solution: 5


Combinatorial Optimization with MaxSATExample Problem

A set of books and a bookshelf.Some books are similar others are not.

Want to place the books onto theshelves:

Similar books→ same shelf.Dissimilar book→ different shelves.Can have empty shelves.As many shelves as books.



LOTR

b1b1b1

The Number Devil

The Northern Lights

b3b3b3

A Short history of Time

b4b4b4

Shelf 1Shelf 2Shelf 3Shelf 4

+ +

+

−

−

+

+

+



b1

b1b1b1

b2

b3

b3b3b3

b4

b4b4b4


+ +

+

−

−

+

+

+



b1

b1b1b1

b2

b3

b3b3b3

b4

b4b4b4


+ +

+

−

−

+

+

+



b1b1

b1b1

b2

b3b3

b3b3

b4b4

b4b4


+ +

+

−−

+

+

+



b1

b1

b1

b1

b2

b3

b3

b3

b3

b4

b4

b4

b4


+ +

+

−

−

+

+

+



b1

b1b1

b1

b2

b3

b3b3

b3

b4

b4b4

b4


+ +

+

−

−

+

+

+


Grouping books using MaxSAT

b1

b1

b2

b2

b3

b3

b4

b4

+ +

+

−

MaxSAT-based solution:

Variables:s1

1, . . . , s41

s11 =True if b1 on shelf 1

Constraints:Full formula:

(s11 ∨ s2

1 ∨ s31 ∨ s4

1) ∧∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2) ∧∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...

(s11 ∨ s2

1 ∨ s31 ∨ s4

1 ∨ s41) ∧

∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2 ∨ s42) ∧

∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...



b1

b1

b2

b2

b3

b3

b4

b4

+ +

+

−

MaxSAT-based solution:1) Encode

Variables:s1

1, . . . , s41



(s11 ∨ s2

1 ∨ s31 ∨ s4

1) ∧∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2) ∧∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...

(s11 ∨ s2

1 ∨ s31 ∨ s4

1 ∨ s41) ∧

∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2 ∨ s42) ∧

∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...



b1

b1

b2

b2

b3

b3

b4

b4

+ +

+

−


Variables:s1

1, . . . , s41



(s11 ∨ s2

1 ∨ s31 ∨ s4

1) ∧∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2) ∧∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...

(s11 ∨ s2

1 ∨ s31 ∨ s4

1 ∨ s41) ∧

∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2 ∨ s42) ∧

∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...



b1

b1

b2

b2

b3

b3

b4

b4

+ +

+

−


Variables:s1

1, . . . , s41


Constraints:Hard: b1 put on exactly one shelf.

(s11 ∨ s2

1 ∨ s31 ∨ s4

1) ∧∧

i 6=j ¬(si1 ∧ sj

1)

Full formula:

(s11 ∨ s2

1 ∨ s31 ∨ s4

1) ∧∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2) ∧∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...

(s11 ∨ s2

1 ∨ s31 ∨ s4

1 ∨ s41) ∧

∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2 ∨ s42) ∧

∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...



b1

b1

b2

b2

b3

b3

b4

b4

+ +

+

−


Variables:s1

1, . . . , s41


Constraints:Soft: prefer b1 and b3 on same shelf.∨4

k=1(sk1 ∧ sk

3)

Full formula:

(s11 ∨ s2

1 ∨ s31 ∨ s4

1) ∧∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2) ∧∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...

(s11 ∨ s2

1 ∨ s31 ∨ s4

1 ∨ s41) ∧

∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2 ∨ s42) ∧

∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...



b1

b1

b2

b2

b3

b3

b4

b4

+ +

+

−


Variables:s1

1, . . . , s41


Constraints:

Full formula:(s1

1 ∨ s21 ∨ s3

1 ∨ s41) ∧

∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2) ∧∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...

(s11 ∨ s2

1 ∨ s31 ∨ s4

1 ∨ s41) ∧

∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2 ∨ s42) ∧

∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...



b1

b1

b2

b2

b3

b3

b4

b4

+ +

+

−

MaxSAT-based solution:1) Encode2) Solve

Variables:s1

1, . . . , s41


Constraints:

Full formula:

(s11 ∨ s2

1 ∨ s31 ∨ s4

1) ∧∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2) ∧∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...

(s11 ∨ s2

1 ∨ s31 ∨ s4

1 ∨ s41) ∧

∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2 ∨ s42) ∧

∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...



b1b1

b2b2

b3b3

b4b4

+ +

+

−

MaxSAT-based solution:1) Encode2) Solve3) Reconstruct

Variables:s1

1, . . . , s41


Constraints:

Full formula:

(s11 ∨ s2

1 ∨ s31 ∨ s4

1) ∧∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2) ∧∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...

(s11 ∨ s2

1 ∨ s31 ∨ s4

1 ∨ s41) ∧

∧i 6=j

¬(si1 ∧ sj

1) ∧ (s12 ∨ s2

2 ∨ s32 ∨ s4

2 ∨ s42) ∧

∧i 6=j

¬(si2 ∧ sj

2) ∧ . . .

4∨k=1

(sk1 ∧ sk

3 ) ∧4∨

k=1

(sk1 ∧ sk

4 ) ∧4∨

k=1

(sk1 ∧ sk

2 ) ∧4∧

k=1

¬(sk3 ∧ sk

4 ) ∧ . . .

...


Declarative Approach to Combinatorial OptimizationMore generally

Instance p of P MaxSAT instance F(p)

Solution τ to F(p)Solution to p

MaxSAT encoding

MaxSAT solver

Reconstruction

Benefits of the declarative approachGeneralFlexibleCompetitive with specialized algorithms





MaxSAT encoding

MaxSAT solver

Reconstruction






MaxSAT encoding

MaxSAT solver

Reconstruction






MaxSAT encoding

MaxSAT solver

Reconstruction



Thesis researchIn more detail



MaxSAT encoding

MaxSAT solver

Reconstruction

Six peer reviewed publications aiming to improve MaxSAT-basedsolution methods to combinatorial optimization problems.

Part 1: MaxSAT solving techniques, specifically preprocessing(Papers I - IV)Part 2: MaxSAT encodings of other data analysis problems(Papers V - VI)





MaxSAT encoding

MaxSAT solver

Reconstruction







MaxSAT encoding

MaxSAT solver

Reconstruction




Part 1: Preprocessing

Instance p of P Instance F(p)

Preprocessed (Re-encoded)instance pre(F(p))

Solution τto pre(F(p))

Solution to p

Encoding

Preprocessor

Solver

Reconstruction+

Simplification rules that re-encode MaxSAT instances.Decrease overall solving time

In practice, beneficial on several instances.


Contributions to MaxSAT Preprocessing

1 Efficient integration of preprocessing to modern core-guidedMaxSAT solving (Papers I & II)

Improving empirical performance of especially the implicit hitting setbased solvers

2 Theoretical analysis on the effects of preprocessing oncore-guided MaxSAT solving (Paper III)

Preprocessing has limited effect on the number of iterationsrequired by core-guided solvers.

3 A new preprocessing rule for MaxSAT (Paper IV)Subsumed Label Elimination


Example:

b1

b1

b2

b3

b3

b4

b4b4

+ +

+

−

+ −

Subsumed Label Elimination (SLE):

–Enforce b1 and b4 onto the same shelf during preprocessing.–SLE captures a subset of such simplifications on the MaxSAT level.


Example:

b1b1

b2

b3b3

b4b4

b4

+ +

+

−+

−

Subsumed Label Elimination (SLE):–Enforce b1 and b4 onto the same shelf during preprocessing.

–SLE captures a subset of such simplifications on the MaxSAT level.


Example:

b1b1

b2

b3b3

b4

b4

b4

+ +

+

−

+

−

Subsumed Label Elimination (SLE):–Enforce b1 and b4 onto the same shelf during preprocessing.

–SLE captures a subset of such simplifications on the MaxSAT level.


Example:

b1b1

b2

b3b3

b4

b4

b4

+ +

+

−

+ −

Subsumed Label Elimination (SLE):–Enforce b1 and b4 onto the same shelf during preprocessing.–SLE captures a subset of such simplifications on the MaxSAT level.


Part 2: MaxSAT encodings of other problems

Contributions:1 Propose new MaxSAT encodings of two data analysis problems

Correlation Clustering (Paper V)Bounded Treewidth Bayesian Network Structure Learning(Paper VI)

2 Experimental evaluation comparing MaxSAT based solutionmethods with previously proposed solution algorithms.


Correlation Clustering

Generalization of ”the book placing problem”.Cluster a set of datapoints based on pairwise similarityinformation.Dual natured objective function:

Maximize sum of weights of positive edges assigned to the samecluster.Minimize sum of weights of negative edges assigned to the samecluster.

Applications in: biosciences, social network analysis, informationretrieval, bioinformatics and microarray data analysis . . . .


Correlation ClusteringExperiments on sparse data

x1

x2

x3

x4

x5

Make sparse instances bykeeping p ∗ 100% of the edges

p = 1.0

Compute cost w.r.t to full instance

+

+

− +

−

−

+

+−

−



x1

x2

x3

x4

x5


p = 0.7


+

+

−

−

+

−

−



x1

x2

x3

x4

x5


p = 0.4


+

+

−−



x1

x2

x3

x4

x5


p = 0.1


+


Experiments on Sparse DataResults

600

700

800

900 1000

1200

1500

2100

2700

3500

4500

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Co

st

of

Clu

ste

rin

g

p

SDPCKC

SCPSCCA

MaxSAT-Binary


Bayesian Networks


Bounded Treewidth Bayesian Network StructureLearning (BTBNSL)

BNSL: Learn a network structure which ”explains” a set ofobservations (data) as well as possible.Do probabilistic inference over the network

Both are NP-hard in general

BTBNSL: Learn a network over which inference is effective.

A S T L B T&L X D0 1 0 1 0 1 1 10 0 1 1 0 1 0 11 1 0 1 1 0 0 11 0 0 1 1 1 1 11 1 1 1 1 1 0 11 1 1 1 0 0 1 0...

......

......

......

...

P(S | T , L,B) =?

P(X | D,A,B) =?

P(A | T , L,B) =?...


Bounded Treewidth Bayesian Network StructureLearning (BTBNSL)

BNSL: Learn a network structure which ”explains” a set ofobservations (data) as well as possible.Do probabilistic inference over the network

Both are NP-hard in general

BTBNSL: Learn a network over which inference is effective.

A S T L B T&L X D0 1 0 1 0 1 1 10 0 1 1 0 1 0 11 1 0 1 1 0 0 11 0 0 1 1 1 1 11 1 1 1 1 1 0 11 1 1 1 0 0 1 0...

......

......

......

...

P(S | T , L,B) =?

P(X | D,A,B) =?

P(A | T , L,B) =?...


Treewidth

x1

x2

x3

x4

x5

Treewidth: 1

x1

x2

x3

x4

x5

Treewidth: 2

x1

x2

x3

x4

x5

Treewidth: 4

Treewidth of a graph measures its distance from being a treeBayesian inference is effective over structures with boundedtreewidth


BTBNSLComparison of exact approaches

0

5000

10000

15000

20000

25000

10 20 30 40 50 60

Tim

eout (s

)

Instances solved

MaxSATDPILP


Conclusions

In this thesis we:Propose and evaluate a method of integrating preprocessing intocore-guided MaxSAT solving.Analyze the effect of preprocessing on core-guided MaxSATsolving.Propose and evaluate a new preprocessing rule for MaxSAT.Propose and evaluate MaxSAT encodings of two data analysisproblems.

Improve the organization of bookshelves everywhere (hopefully).


Conclusions




Conclusions




Conclusions




Conclusions




Conclusions

In this thesis we:Propose and evaluate a method of integrating preprocessing intocore-guided MaxSAT solving.Analyze the effect of preprocessing on core-guided MaxSATsolving.Propose and evaluate a new preprocessing rule for MaxSAT.Propose and evaluate MaxSAT encodings of two data analysisproblems.Improve the organization of bookshelves everywhere (hopefully).


Documents

Solving Optimization Problems via Maximum Satisfiability ... · Lectio Praecursoria Jeremias Berg J. Berg Lectio praecursoria 1 / 22. Thesis research Contributions to exact declarative