The document discusses algorithms for solving maximum satisfiability (MaxSAT) problems. It presents the unit propagation (UP) algorithm, which can detect inconsistent subformulas. An example MaxSAT instance is given and solved step-by-step using UP. The document also discusses failed literal detection, variable selection strategies, and data structures that can improve the efficiency of MaxSAT solving. Variable ordering is described based on calculating weights using lengths, counts of binary, unit and total clauses.

1. By: Mohammad Khosravi , Reza Ramezani
To: Dr Moosavi
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Electronic & Computer Department
Isfahan University Of Technology

2.  Input:
given a list of clauses C1, . . . ,Cm
 Output:
the maximum number of clauses which
satisfied.
 Unit Propagation (UP) algorithm use for
compute LB when we have at least one unit
clause, otherwise we will use Failed literal
detection.
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3.  Let 1 be the Max-SAT instance
{x1, x2, x3,¬x1 x4,¬x1 x5,¬x4 ¬x5,¬x1 ¬x2 ¬x3, x1 ¬x2}
Q = [x1, x2, x3]
 x1 propagate
{ x2, x3, x4, x5,¬x4 ¬x5, ¬x2 ¬x3}
Q = [x2,x3,x4,x5]
 x2 propagate
{x3, x4, x5,¬x4 ¬x5,¬x3}
Q =[x3, x4, x5,¬x3]
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4.  x3 propagate
{#,x4, x5,¬x4 ¬x5}
Inconsistent subformula detected
by UP is {x1, x2, x3,¬x1 ¬x2 ¬x3}
The remaining clauses {¬x1 x4,¬x1 x5,¬x4 ¬x5, x1 ¬x2}
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5. {x1, x2, x3,¬x1 x4,¬x1 x5,¬x4 ¬x5,¬x1 ¬x2 ¬x3, x1
¬x2}
Q1 = [x1, x2, x3]
 x1 propagate
Q1 = [ x2, x3]
{ x2, x3, x4, x5,¬x4 ¬x5, ¬x2 ¬x3}
Q2 = [x4, x5]
 x4 propagate
Q2 = [x5,¬x5]
 x5 propagate
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6.  Inconsistent subformula : {x1,¬x1 x4,¬x1 x5,¬x4
¬x5}
 remaining clauses:
{x2, x3,¬x1 ¬x2 ¬x3, x1 ¬x2}
Q1 = [x2, x3]
 x2 propagate
{x3,¬x1 ¬x3, x1 }
Q1 = [x3]
Q2 = [x1]
 x1 propagate
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7. {x3, ¬x3 }
Q2 = [¬x3 ]
 inconsistent subformula : {x2, x3,¬x1 ¬x2 ¬x3, x1 ¬x2}.
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8. P={#,#, y ¬x,¬y z,¬y ¬z, y x}
 P U {x},with UP
 inconsistent subformula 1 = {x, y ¬x,¬y z,¬y ¬z},
P U {¬ x}, with UP inconsistent
 Subformula 2= {¬x,¬y z,¬y ¬z, y x}
 Final incosistant subformula=(1 U 2) {x,¬x} = {y
¬x,¬y z,¬y ¬z, y x}
 We can use FL whit UP & UP*
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9. Solutions for solving maxsat efficiency:
Rules
Unit Propagation
Failed Literal Detection
Selecting variables for propagation
Data Structure
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10. Each variable order computed from:
 Binary-Length with two value.
 Unit-Length with one value.
 Total-Length that contain the number of clauses that
variable occurs.
 W(i) is sum of all above weights.
 This weight will calculate for Var(i) and Var(-i).
 Final Variable’s weight is W(i) + W(-i)
 Each Variable with higher weight with select earlier for
propagation
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11. Each variable order computed from:
 Binary-Length with three value.
 Unit-Length with two value.
 Third-Length with one value.
 Total-Length that contain the number of clauses that
variable occurs.
AKMax-Sat solver ReCompute this weights at each
level, such as: propagate variable, remove variable and
restore variable.
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