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Student: Victoria Kravchenko Supervisors: P rof. Yoram Moses Liat Atsmon. Sat solving maximization and minimization problems. The Project Goal. Study and evaluate the approach of solving optimization problems through SAT reduction, and using SAT solvers to solve the reduced problems. SAT.

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Student victoria kravchenko supervisors p rof yoram moses liat atsmon

Student: Victoria Kravchenko

Supervisors: Prof. Yoram Moses

LiatAtsmon

Sat solving maximization and minimization problems


The project goal
The Project Goal

Study and evaluate the approach of solving optimization problems through SAT reduction, and using SAT solvers to solve the reduced problems.


SAT

  • Boolean satisfiability problem

    (x1+x5’+x4) (x1’+x5+x3+x4) (x3’+x4’)

  • NP-Complete

  • CNF (Conjunctive Normal Form) – “AND of ORs”


Minisat
MiniSat

  • Developed at 2003 byNiklasEén and NiklasSörensson

  • Advantages: open-source, successful, small.


The minisat input form
The MiniSat input form

For: (x1+x5’+x4) (x1’+x5+x3+x4) (x3’+x4’)

p cnf 5 3 → p cnfnumber_of_variablesnumber_of_clauses

1 -5 4 0 → 0 is the end of the clause

-1 5 3 4 0

-3 -4 0


The project
The Project

  • PART 1: Equation Checker

  • PART 2: SAT with Optimization

  • PART 3: Maximal Acyclic Subgraph


The equation checker
The equation checker

  • Input: X, Y, Z – decimal numbers

  • Output: whether X+Y=Z or not

  • Calculates the binary form

    of the numbers

  • The program preparesthe input for MiniSatusing the table:


The equation checker1
The equation checker

  • For the LSB bit:



Sat with optimization
SAT with Optimization

  • Input: Vec – x0,x1,x2,…

    Phi – CNF expression

  • Output: maximal value of Vec (as a binary number) which satisfies Phi


Sat with optimization1
SAT with Optimization

  • Versions:

  • Bit by Bit:

  • MiniSat condition:

MSB

1?

1

1?

0

1?

1

?1

?

1

?

0

1

1

?

0

0

?

1

0

?

1


Maximal acyclic subgraph
Maximal Acyclic Subgraph

  • Input: graph name nodes – [node_number],[node_weight] edges – [node_from],[node_to]

    For example: ex1 1,2-2,2-3,1 1,2-2,3-3,1

1(2)

2(2)

3(1)


Maximal acyclic subgraph1
Maximal Acyclic Subgraph

  • Output: maximal acyclic subgraph

1(2)

2(2)

3(1)


Example 1
Example 1

P

1 (1)

2 (2)

3 (2)

4 (3)

5 (3)

lsg ex1 1,1-2,2-3,2-4,3-5,3 1,2-1,3-2,3-2,4-4,5-5,3


Example 2
Example 2

1 (1)

2 (2)

3 (2)

4 (3)

5 (3)

lsg ex2 1,1-2,2-3,2-4,3-5,3 1,2-3,1-2,3-2,4-4,5-5,3


Example 3
Example 3

1 (2)

2 (1)

3 (2)

4 (3)

5 (3)

lsgex3 1,2-2,1-3,2-4,3-5,3 1,2-3,1-2,3-2,4-4,5-5,3


Example 4
Example 4

1 (1)

2 (2)

3 (2)

4 (3)

5 (3)

lsgex4 1,1-2,2-3,2-4,3-5,3 1,2-3,1-2,3-2,4-4,5-5,2-5,3


Example 5
Example 5

1 (1)

2 (4)

3 (2)

4 (3)

5(1)

lsgex5 1,1-2,4-3,2-4,3-5,1 1,2-3,1-2,3-2,4-4,5-5,2-5,3


How does it work
How Does It Work?

For each edge:

  • Define: [node_from] > [node_to]

  • Condition:

  • Translate the condition to CNF using a tree

    (reduction to 3-SAT)


How does it work1
How Does It Work?

Building the weighted sum:

  • [node] x [weight]:

    Y1[m vars] ↔ X1[node number] x W1[m vars]

    Y2[m vars] ↔ X2[node number] x W2[m vars]

  • Sum the intermediate variables as in the Equation Checker: Z1[m vars]↔ Y1[m vars] + Y2[m vars]

  • Continue summing: Z2[m vars]↔ Z1[m vars] + Y3[m vars]

  • Save the last intermediate variables representing the accumulated sum: Z(n-1) [m vars] ↔ Z(n-2) [m vars] + Yn[m vars]

  • Zfinal↔ X1*W1 + X2*W2 + … + Xn*Wn


How does it work2
How Does It Work?

  • Get a result from the MiniSat using only the edges conditions and extract the weight Wres

  • Demand a new result with a greater weight s.t. Zfinal > Wres

  • Continue until the conditions can not be saturated






Conclusions
Conclusions

  • The graphs are polynomial and not exponential although the problem in NP complete – success!

  • Nodes have a stronger effect on the run time.



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