Message Passing and Local Heuristics as Decimation Strategies for Satisfiability - PowerPoint PPT Presentation

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Message Passing and Local Heuristics as Decimation Strategies for Satisfiability

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make local modifications ('flips') to a candidate assignment. until a solution is found ... backtracking and 'flipping' values as necessary ... – PowerPoint PPT presentation

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Title: Message Passing and Local Heuristics as Decimation Strategies for Satisfiability


1
Message Passing and Local Heuristicsas
Decimation Strategies for Satisfiability
  • Lukas Kroc, Ashish Sabharwal, Bart Selman
  • (presented by Sebastian Brand)
  • Symposium on Applied ComputingMarch 2009

2
Combinatorial Search Procedures
  • Search procedures for combinatorial problems such
    as SATusually fall into two categories
  • Local search make local
    modifications ('flips') to a candidate
    assignment until a solution is found
  • Systematic backtrack search
  • explore the search space through partial
    assignment, backtracking and
    'flipping' values as necessary
  • Decimation is a third mechanism, which has
    recently shown tremendous success on hard classes
    of SAT instances

the focus of this paper
3
A Study of Decimation-BasedSatisfiability
Algorithms
  • What is decimation?
  • Natural strategies or heuristics for
    decimation?
  • simple 'local' heuristics
  • message passing 'global' heuristics,
    specifically, belief propagation survey
    propagation
  • How far can these heuristics push decimation?
  • survey propagation extremely successful on
    random k-SAT
  • What makes survey propagation different?
  • Need measurable properties that highlight
    differences.
  • evolution of problem 'hardness' during
    decimation
  • generation (or not) of unit clauses

4
Talk Outline
  • The decimation process (for solving SAT)
  • Decimation strategies
  • Local heuristics
  • Global message passing heuristics
  • Empirical comparison
  • Differences in decimation strategies

5
The Decimation Procedure
  • Given some ordering of the variable-value pairs
  • Do
  • Assign the first variable its value
  • Simplify the problem instance
  • Recompute the ordering and repeat
  • Very scalable!
  • No repair mechanism ? the ordering must be
    smart to eventually find a solution

Where do we get the smart ordering from?
6
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9
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10
Belief Propagation for Inference
  • The original BP does not converge
  • first need to dampen it to force convergence

Damping constant1 same as BP 0 guaranteed
convergence
11
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12
Using Surveys Instead of Beliefs
  • BP-inspired-decimation does not work for very
    hard random instances

SAT instance
Solve it bydecimation
Use BP forPrxTsolution
  • For more difficult random SAT problems, use
    SP-inspired-decimation
  • Modify the problem itself

Use SP forPrxTcover
SAT instance
Solve it bydecimation
13
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14
First experimental study How well do various
decimation heuristics perform?
15
Results How Far Does Decimation Go?
LUKAS please put a couple of key points here,
that these bar plots bring out.Mention what we
are comparing.
16
Second experimental study What are some
measurable properties that provideinsights into
Survey Propagation vs. other heuristics?
Note common measures such as number of 2- and
3-clauses, or positive vs. negative
literals, etc., do not show any measurable
difference
17
Generation of Unit Clauses
  • Unlike all other heuristics considered,SP
    generates nearly no unit propagationsuntil
    around 40 of the variables are set!

18
Generation of Unit Clauses
  • PropositionIf the computed marginals (solution
    or cover) are perfect and the maximum
    magnetization is unique, then there will be no
    unit propagation at all.
  • SP's computation is, indeed, close to perfect, at
    least in the extreme magnetization regions.

19
Evolution of Problem Hardness
  • Measure hardness of the residual formula at
    every step as no. of flips Walksat needs to find
    solution
  • Unlike all other heuristics considered, SP
    constantly reduces the hardness of the residual
    formula!

20
Summary
  • Global decimation heuristics, based on message
    passing, are much more effective than local ones
  • SP is much more accurate in computing marginal
    estimates than BP (on hard random instances)
  • SP shows two unique characteristics as decimation
    evolves
  • Nearly no unit propagations generated
  • Instance constantly becomes easier
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