Local Search for Satisfiability

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Local Search for Satisfiability
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CNF Satisfiability Problem

• Input. Logical formula in conjunctive normal form
  (AND of OR-clauses) (x1 or not(x2) or x3) and
  (x2 or x3 or not(x4)) and (not(x1) or x2 or x4)
• Output. Satisfying boolean assignment
• k-CNF SAT. Each OR-clause contains exactly k
  literals)
• 3-CNF SAT. Each OR-clause contains exactly 3
  literals
• All of these problems are NP-Complete.
• Can randomized search help solve them?

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Simple Random Walk for CNF Satisfiability

• Simple Random Walk Start with a random truth
  assignment. Pick a random unsatisfied clause and
  flip the truth value of a random variable in that
  clause. Repeat until formula satisfied.
• Simple Random Walk is good enough for 2-CNF
  formulas Solves any satisfiable instance in
  expected O(n2) time Papadimitriou 1991.
• Simple Random Walk is bad for 3-CNF formulas
  Some satisfiable instances require exponential
  expected time Papadimitriou 1994. Also
  empirically bad on random 3-CNF formulas.

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Local Search for Satisfiability

• Construct a cost function for truth assignments
  that is minimized when the formula is satisfied.
• Guess a truth assignment (e.g., at random).
• Try to improve the guess by checking nearby
  truth assignments for one with lower cost
  (repeatedly).

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Hill-climbing search
Thanks to Min-Yen Kan

• Problem depending on initial state, can get
  stuck in local maxima

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Local Search for CNF Satisfiability

• cost function for truth assignments number of
  satisfied clauses
• nearest truth assignments flip the value of
  one boolean variable
• Example x1 1, x2 0, x3 0, x4 1 has cost
  2 for CNF formula
• (x1 or not(x2) or x3) and (x2 or x3 or not(x4))
  and (not(x1) or x2 or x4).
• Nearest truth assignments x1 0, x2 0, x3
  0, x4 1 has cost 2
• x1
  1, x2 1, x3 0, x4 1 has cost 3
• x1
  1, x2 0, x3 1, x4 1 has cost 3
• x1
  1, x2 0, x3 0, x4 0 has cost 2

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GSAT Algorithm (Selman et al. 1992)

• GSAT(k-SAT formula with variables x_1, , x_n)
• Repeat MAX-TRIES times (or until formula is
  satisfied)
• Let A be a random boolean assignment for x_1,
  , x_n
• Repeat MAX-FLIPS times (or until formula is
  satisfied)
• For i 1 to n
• Let Ni number of clauses satisfied
  by A if x_i value flipped
• Let i be the index that maximizes Ni
• Flip the value of x_i in A
• NOTE The flip is made even if the best flip
  increases the number of unsatisfied clauses. If
  GSAT only makes improving steps, then it
  performs very poorly.

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Iterative Repair

• Unsatisfied clauses symptoms of a flaw in
  the truth assignment that needs to be repaired
• Different from local search Every step repairs a
  flaw but might make the natural cost function
  worse.
• Simple Random Walk is an example of Iterative
  Repair.

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Iteraritive Repair WALKSAT (Selman et al. 1994)

• WALKSAT(k-SAT formula with variables x_1, , x_n)
• Repeat MAX-TRIES times (or until formula is
  satisfied)
• Let A be a random boolean assignment for x_1,
  , x_n
• Repeat MAX-FLIPS times (or until formula is
  satisfied)
• Choose a random clause c that is not
  satisfied under assignment A
• Choose a variable x in c (randomly or to
  max of sat clauses)
• Flip the value of x in A
• This Iterative Repair algorithm performs very
  well in practice.

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Iterative Repair Novelty (McAllester et al. 1997)

• Novelty(k-SAT formula with variables x_1, , x_n)
• Repeat MAX-TRIES times (or until formula is
  satisfied)
• Let A be a random boolean assignment for x_1,
  , x_n
• Repeat MAX-FLIPS times (or until formula is
  satisfied)
• Choose a random clause c that is not
  satisfied under assignment A
• Choose a variable x in c greedily (to max
  of satisfied clauses)
• If x was not the most recently flipped
  variable in clause c
• then flip its value in A
• If x was the most recently flipped
  variable in clause c
• then with probability 1-p flip its
  value in A anyway
• else flip the value in A of the
  second-best variable in c
• This Iterative Repair algorithm performs well in
  practice.

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Simulated Annealing for Satisfiability

• Basic Simulated Annealing Algorithm(k-SAT
  formula)
• Let temperature T Initial-Temperature
• Let A be a random boolean assignment for x_1, ,
  x_n
• For i 1 to MAX-STEPS do
• Pick a random variable x
• Let d change in the number of unsatisfied
  clauses if x flipped
• If d lt 0, then flip x
• If d gt 0, then flip x with probability e-d/T
• Reduce the temperature T
• The low-probability flips let you escape from a
  local minimum. Its easier to escape early on
  when the temperature is higher.

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Metropolis Algorithm
Simulated Annealing Gradually decrease T