Title: Message Passing and Local Heuristics as Decimation Strategies for Satisfiability
1Message Passing and Local Heuristicsas
Decimation Strategies for Satisfiability
- Lukas Kroc, Ashish Sabharwal, Bart Selman
- (presented by Sebastian Brand)
- Symposium on Applied ComputingMarch 2009
2Combinatorial 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
3A 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
4Talk Outline
- The decimation process (for solving SAT)
- Decimation strategies
- Local heuristics
- Global message passing heuristics
- Empirical comparison
- Differences in decimation strategies
5The 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?
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10Belief 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
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12Using 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
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14First experimental study How well do various
decimation heuristics perform?
15Results 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.
16Second 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
17Generation of Unit Clauses
- Unlike all other heuristics considered,SP
generates nearly no unit propagationsuntil
around 40 of the variables are set!
18Generation 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.
19Evolution 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!
20Summary
- 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