Local Search Techniques Marijn J.H. Heule Warren A. Hunt Jr. The - PowerPoint PPT Presentation

Local Search Techniques Marijn J.H. Heule Warren A. Hunt Jr. The University of Texas at Austin Heule & Hunt (UT Austin) Local Search Techniques 1 / 8

Local Search: Generic structure Generic structure of local search Sat solvers 1: for i in 1 to MAX TRIES do ϕ := random initial assignment 2: for j in 1 to MAX STEPS do 3: if ϕ satisfies F then 4: return satisfiable 5: end if 6: ϕ := Flip ( ϕ ) 7: end for 8: 9: end for 10: return unknown Heule & Hunt (UT Austin) Local Search Techniques 2 / 8

Local Search: Global vs Local flips Global flips Pro: Big improvements Neg: Probabilistic incomplete Local flips Neg: Small improvements Pos: Probabilistic complete Heule & Hunt (UT Austin) Local Search Techniques 3 / 8

Local Search: WalkSAT Flips Select a random unsatisfied clause C Free flip Random flip Heuristic flip Heule & Hunt (UT Austin) Local Search Techniques 4 / 8

Local Search: WalkSAT Code Flip WalkSat ( ϕ ) 1: C := random falsified clause by ϕ ◦ F 2: if a variable ∈ C can be flipped for free then flip in ϕ that variable 3: 4: else flip in ϕ with p a random x i ∈ C 5: flip in ϕ with 1 − p the “optimal” x i ∈ C 6: 7: end if 8: return ϕ Heule & Hunt (UT Austin) Local Search Techniques 5 / 8

The UnitWalk Algorithm π 0 π 1 π 2 UnitWalk UnitWalk UnitWalk . . . period period period ϕ 0 ϕ 1 ϕ 2 ϕ 3 Heule & Hunt (UT Austin) Local Search Techniques 6 / 8

The UnitWalk Algorithm π 0 π 1 π 2 UnitWalk UnitWalk UnitWalk . . . period period period ϕ 0 ϕ 1 ϕ 2 ϕ 3 The general idea of a UnitWalk period: Within each unsatisfied clause in ϕ i ◦ F the assignment to the least important variable (based on π i ) is flipped Heule & Hunt (UT Austin) Local Search Techniques 6 / 8

The UnitWalk Algorithm π 0 π 1 π 2 UnitWalk UnitWalk UnitWalk . . . period period period ϕ 0 ϕ 1 ϕ 2 ϕ 3 The general idea of a UnitWalk period: Within each unsatisfied clause in ϕ i ◦ F the assignment to the least important variable (based on π i ) is flipped For example: F = ( x ∨ ¬ y ) , ϕ 0 = { x = 0 , y = 1 } , π 0 = { y , x } Heule & Hunt (UT Austin) Local Search Techniques 6 / 8

UnitWalk Period Example F example := ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ∨ x 3 ) ∧ ( ¬ x 2 ∨ ¬ x 3 ) ∧ ( ¬ x 2 ∨ x 3 ∨ ¬ x 4 ) ∧ ( ¬ x 2 ∨ x 3 ∨ x 4 ) ∧ ( ¬ x 3 ∨ ¬ x 4 ) := { x 1 = 0 , x 2 = 1 , x 3 = 1 , x 4 = 0 } ϕ master := { x 1 = ∗ , x 2 = ∗ , x 3 = ∗ , x 4 = ∗} ϕ active := ( x 2 , x 1 , x 4 , x 3 ) π do iterative propagate unit clauses in ϕ active ◦ F example extend ϕ active with most important free variable according to π while ϕ active contains ∗ ’s Heule & Hunt (UT Austin) Local Search Techniques 7 / 8

UnitWalk Period Example F example := ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ∨ x 3 ) ∧ ( ¬ x 2 ∨ ¬ x 3 ) ∧ ( ¬ x 2 ∨ x 3 ∨ ¬ x 4 ) ∧ ( ¬ x 2 ∨ x 3 ∨ x 4 ) ∧ ( ¬ x 3 ∨ ¬ x 4 ) := { x 1 = 0 , x 2 = 1 , x 3 = 1 , x 4 = 0 } ϕ master := { x 1 = ∗ , x 2 = ∗ , x 3 = ∗ , x 4 = ∗} ϕ active := ( x 2 , x 1 , x 4 , x 3 ) π do → iterative propagate unit clauses in ϕ active ◦ F example extend ϕ active with most important free variable according to π while ϕ active contains ∗ ’s Action: no unit clauses in ϕ active ◦ F example Heule & Hunt (UT Austin) Local Search Techniques 7 / 8

UnitWalk Period Example F example := ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ∨ x 3 ) ∧ ( ¬ x 2 ∨ ¬ x 3 ) ∧ ( ¬ x 2 ∨ x 3 ∨ ¬ x 4 ) ∧ ( ¬ x 2 ∨ x 3 ∨ x 4 ) ∧ ( ¬ x 3 ∨ ¬ x 4 ) := { x 1 = 0 , x 2 = 1 , x 3 = 1 , x 4 = 0 } ϕ master := { x 1 = ∗ , x 2 = 1 , x 3 = ∗ , x 4 = ∗} ϕ active := ( x 2 , x 1 , x 4 , x 3 ) π do iterative propagate unit clauses in ϕ active ◦ F example → extend ϕ active with most important free variable according to π while ϕ active contains ∗ ’s Action: extend ϕ active with x 2 := 1 (the truth value in ϕ master ) Heule & Hunt (UT Austin) Local Search Techniques 7 / 8

UnitWalk Period Example F example := ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ∨ x 3 ) ∧ ( ¬ x 2 ∨ ¬ x 3 ) ∧ ( ¬ x 2 ∨ x 3 ∨ ¬ x 4 ) ∧ ( ¬ x 2 ∨ x 3 ∨ x 4 ) ∧ ( ¬ x 3 ∨ ¬ x 4 ) := { x 1 = 0 , x 2 = 1 , x 3 = 1 , x 4 = 0 } ϕ master := { x 1 = ∗ , x 2 = 1 , x 3 = 0 , x 4 = ∗} ϕ active := ( x 2 , x 1 , x 4 , x 3 ) π do → iterative propagate unit clauses in ϕ active ◦ F example extend ϕ active with most important free variable according to π while ϕ active contains ∗ ’s Action: detected unit clause ¬ x 3 → x 3 := 0 Heule & Hunt (UT Austin) Local Search Techniques 7 / 8

UnitWalk Period Example F example := ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ∨ x 3 ) ∧ ( ¬ x 2 ∨ ¬ x 3 ) ∧ ( ¬ x 2 ∨ x 3 ∨ ¬ x 4 ) ∧ ( ¬ x 2 ∨ x 3 ∨ x 4 ) ∧ ( ¬ x 3 ∨ ¬ x 4 ) := { x 1 = 0 , x 2 = 1 , x 3 = 1 , x 4 = 0 } ϕ master := { x 1 = ∗ , x 2 = 1 , x 3 = 0 , x 4 = 0 } ϕ active := ( x 2 , x 1 , x 4 , x 3 ) π do → iterative propagate unit clauses in ϕ active ◦ F example extend ϕ active with most important free variable according to π while ϕ active contains ∗ ’s Action: detected unit clauses x 4 and ¬ x 4 → conflict assign x 4 to truth value in ϕ master → x 4 := 0 Heule & Hunt (UT Austin) Local Search Techniques 7 / 8

UnitWalk Period Example F example := ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ∨ x 3 ) ∧ ( ¬ x 2 ∨ ¬ x 3 ) ∧ ( ¬ x 2 ∨ x 3 ∨ ¬ x 4 ) ∧ ( ¬ x 2 ∨ x 3 ∨ x 4 ) ∧ ( ¬ x 3 ∨ ¬ x 4 ) := { x 1 = 0 , x 2 = 1 , x 3 = 1 , x 4 = 0 } ϕ master := { x 1 = 0 , x 2 = 1 , x 3 = 0 , x 4 = 0 } ϕ active := ( x 2 , x 1 , x 4 , x 3 ) π do iterative propagate unit clauses in ϕ active ◦ F example → extend ϕ active with most important free variable according to π while ϕ active contains ∗ ’s Action: extend ϕ active with x 1 := 0 (the truth value in ϕ master ) Heule & Hunt (UT Austin) Local Search Techniques 7 / 8

UnitWalk Period Example F example := ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ∨ x 3 ) ∧ ( ¬ x 2 ∨ ¬ x 3 ) ∧ ( ¬ x 2 ∨ x 3 ∨ ¬ x 4 ) ∧ ( ¬ x 2 ∨ x 3 ∨ x 4 ) ∧ ( ¬ x 3 ∨ ¬ x 4 ) := { x 1 = 0 , x 2 = 1 , x 3 = 1 , x 4 = 0 } ϕ master := { x 1 = 0 , x 2 = 1 , x 3 = 0 , x 4 = 0 } ϕ active := ( x 2 , x 1 , x 4 , x 3 ) π do iterative propagate unit clauses in ϕ active ◦ F example extend ϕ active with most important free variable according to π → while ϕ active contains ∗ ’s Action: end of period because all variables are assigned in ϕ active Heule & Hunt (UT Austin) Local Search Techniques 7 / 8

Local Search Techniques Marijn J.H. Heule Warren A. Hunt Jr. The University of Texas at Austin Heule & Hunt (UT Austin) Local Search Techniques 8 / 8