Why almost all satisfiable kCNF formulas are easy

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Why almost all satisfiable k-CNF formulas are
easy?

• Danny Vilenchik

Joint work with A. Coja-Oghlan and M. Krivelevich
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SAT Basic Notions

• 3CNF form
• F (x1Çx2Çx5) Æ (x3Çx4Çx1) Æ (x1Çx2Çx6) Æ
• Ã
• F ( F ÇF Ç T ) Æ ( T Ç T Ç T ) Æ ( T Ç F Ç T
  )Æ

x5 supports this clause w.r.t. Ã
Goal algorithm that produces optimal result,
efficient, and works for all inputs
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SAT Some Background

• Finding a satisfying assignment is NP Hard
  Cook71
• No approximation for MAX-SAT with factor better
  than 7/8 Hastad01
• How to proceed?
• Hardness results only show that there exist hard
  instances
• The heuristical approach - relaxes the
  universality requirement
• Typical instance?
• One possibility random models

Heuristic is a polynomial time algorithm that
produces optimal results on typical instances
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Random 3SAT

• Random 3SAT
• Fix m,n
• Pick m clauses uniformly at random (over the n
  variables)
• Threshold there exists a constant d such that
  Fri99
• m/nd most 3CNFs are not satisfiable (4.506)
• m/nltd most 3CNFs are satisfiable (3.52)
• Near-threshold 3CNFs are apparently hard for
  many SAT heuristics
• Possible reason complicated structure of
  solution space (clustering)

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Near Threshold Clustering Phenomenon

• Conjectured solution space of Random k-SAT just
  below the threshold
• (part of this picture was rigorously proved for
  k8, AR06,MMZ05)

• All assignments within a
• cluster are close
• A linear number of
• variables are frozen

• Every two clusters are far
• from each other
• Exponentially many clusters

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Our Result

• Rigorously characterize the structure of the
  solution space of Random
• 3SAT, m/n some constant above the threshold

• Single cluster of satisfying assignments
• Size of the cluster is exponential in n
• (1-e-?(m/n))n variables are frozen

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Our Results
Theorem There exists a deterministic polynomial
time algorithm that finds a satisfying
assignment for almost all satisfiable 3CNF
formulas with m/ngtC, C a sufficiently large
constant

• Rigorously complement results for the very sparse
  case
• When clustering is simple the problem is easy
• When clustering is complicated the problem is
  harder (?)
• Improving the exponential time algorithm for
  uniform satisfiable 3CNFs in this regime (only
  one known so far, Chen03)

Almost all k-CNF formulas are easy !
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The Planted Distribution

• Planted 3SAT distribution with parameters m,n
• Fix an assignment ?
• Pick u.a.r. m clauses out of all clauses that
  are satisfied by ?
• Planted 3SAT was analyzed in several papers
• Fla03 shows a spectral algorithm for solving
  sparse instances
• Ben-Sasson et. al. for m/n?(logn) (planted and
  uniform coincide)
• Planted models also fashionable for graph
  coloring, max clique, max independent set, min
  bisection
• Planted models are more approachable clauses
  are practically independent
• Open question how does the planted model compare
  with the uniform?

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Our Result

• We show that the planted and uniform
  distributions share many structural properties
  (close)
• In particular, same structure of the solution
  space
• Justifying the somewhat unnatural usage of
  planted-solution models
• Flaxmans algorithm Fla03 works for the uniform
  distribution as well

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SAT and Message Passing

• FMV06 Warning Propagation was shown to solve
  planted 3SAT instances with m/ngtC, C some
  sufficiently large constant
• Our work implies WP works in the uniform
  setting as well
• Reinforces the following thesis
• When clustering is complicated ) formulas are
  hard ) sophisticated algorithms needed Survey
  Propagation
• When clustering is simple ) formulas are easy )
  naïve algorithms work Warning Propagation

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Clustering Proof Technique

• Recall uniform distribution over satisfiable
  3CNFs with m clauses
• Why more difficult than the planted distribution?
• Edges are not independent
• For starters, consider the planted 3SAT
  distribution
• m/n sufficiently large constant
• Every variable is expected to support 3m/(7n)
  clauses w.r.t. planted
• Prx supports CPrx supports Cx appears in
  CPrx appears in C

Fact 1 whp there is no subformula H on h
variables s.t. hltn/100 and there are at
least hm/(10n) clauses containing two
variables from H
Fact 2 whp there are no two satisfying
assignments at distance greater than n/100
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Clustering Proof Technique
Claim suppose that every variable has the
expected support, and Facts 1 and 2 hold, then F
is uniquely satisfiable

• Proof suppose not,
• Let ? be the planted assignment and à some other
  satisfying assignment
• Take x s.t. Ã(x)??(x), x supports 3m/(7n)
  clauses w.r.t. ?
• Consdier such clause (T Ç F Ç F)
• Define H x Ã(x)??(x) , hHltn/100 (Fact 1)
• There exists 3hm/(7n) clauses containing two
  variables from H
• This contradicts Fact 2.

F
T
Ã
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Clustering Proof Technique

• This picture is whp the case when m/ngtClog n
• When m/nO(1) - whp not the case (some variables
  have 0 support)

Definition Given a 3CNF F and a satisfying
assignment Ã, a set C is called a core of F if
8x2C, x supports at least m/(4n) clauses in FC

• Claim For F in the planted distribution, m/n
  sufficiently large constant
• there exists a core C s.t.
• V(C)gt(1-e-?(m/n))n
• C is frozen in F

Corollary one-cluster structure
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Moving to the Uniform Case

• A a bad structural property (in our case no
  big core)
• ? expected number of satisfying assignments of
  planted 3CNF

Claim PruniformA lt ?PrplantedA
Claim Pruniformno big core lt ?Prplantedno
big corelt ¹e-nc
Claim ¹ltenc, cltc
Corollary Pruniformno big core o(1)
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Further Research
solution space

m/n