Comparing Beliefs, Surveys, and Random Walks for 3SAT

1
Comparing Beliefs, Surveys, and Random Walks for
3-SAT

• Scott Kirkpatrick, Hebrew University
• Joint work with Erik Aurell and Uri Gordon
• (see cond-mat/0406217 v1 9 June 2004)

2
Main Results

• Rederive SP as a special case of BP
• Permits interesting generalizations
• Visualize decimation guided by SP as a flow
• Study the depth of decimation achieved
• WSAT as a measure of formula complexity
• Depends on details of tricks employed
• Shows SP produces renormalization out of hard-SAT
• With todays codes, WSAT outperforms SP!
• Except in regime where 1-RSB is stable
• WSAT has an endpoint at 4.15

3
Beliefs and Surveys

• BP evaluate the probability that variable x is
  TRUE in a solution
• SP evaluate the probability that variable x is
  TRUE in all solutions
• This leaves a third case, x is free to be
  sometimes TRUE sometimes FALSE

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Transports and Influences

• To calculate the beliefs, or the slightly more
  complicated surveys, we introduce quantities
  associated with the directed links of the
  hypergraph
• (transport) T(i?a) fraction of solutions s.t.
  variable i satisfies clause a
• (influence) I(a?i) fraction of solutions s.t.
  clause a is satisfied by variables other than I
• I(a?i) ? T(j?a) T(k?a) T(j?a)T(k?a)
• Same iteration for BP, SP

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Closing the loop introduces a one parameter
family of belief schemes

• Calculate new Ts from the Is, and normalize
• (PPT is equation-challenged do this on the
  board)
• Iterative equations for SP differ from BP in one
  term
• Interpolation formula seems useful in between
• Rho 0 BP
• Rho 1 SP
• 0 lt Rho lt 1 BP ? SP
• 1 lt Rho SP ? unknown
• Interpret effects of Rho in flow diagram

6
Visualize decimation as flows in the SP space
Decimate variables closest to the corners
Origin is the paramagnetic phase
7
BP, SP, hybrids differ in their depth of
decimation
These results are for SP only
8
Depth of decimation achieved by BP, hybrids
9
What is accomplished by decimation?

• A form of renormalization transform
• Simplify the formula by eliminating variables,
  moving out of the hard-SAT regime
• 3.92 lt alpha lt 4.267
• We use WSAT (from H. Kautz, B. Selman, B. Cohen)
  as a standard measure of complexity

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Results of SP decimation
Upper curves WSAT cost/spin Lower curves WSAT
cost/spin after decimation (two normalizations)
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Where does this pay off?

• Using todays programs, with local updates to
  recalculate surveys after each decimation step
• N 10,000, alpha 4.1, 100 formulas
• WSAT only 9.2 sec each
• WSAT after decimation 0.3 sec each
• But SP cost 62 sec each
• N 10,000, alpha 4.2, 100 formulas
• WSAT only 278 sec each
• WSAT after decimation 3 sec each
• SP cost 101 sec each

12
Investigate WSAT more carefully

• WSAT evolved by trial and error, not subject to
  any physical prejudices or intuitions
• Central trick is to always choose an unsat clause
  at random
• Totally focussed on break count number of sat
  clauses which depend on the spin chosen, become
  unsat
• WSAT has one trick not included in the Weigt,
  Monasson studies
• Always check first for free moves, those with
  zero breakcount
• If no free moves, then take random or greedy move
  with equal probability

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Cost per spin is well-defined (linear)
14
WSAT cost/spin variance shrinks with N
Examination of distributions shows that cost/spin
is concentrated as N ? infty up to alpha 4.15!
15
Cumulative distribution of cost/spin alpha 3.9
16
Cumulative distribution of cost/spinalpha 4.1
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Cumulative distribution of cost/spinalpha 4.15
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Cumulative distribution of cost/spin alpha 4.18
19
SP, like rule-based decimation, has an end-point
20
Conclusions

• SP a special case of BP
• Permits interesting generalizations
• Visualize decimation guided by SP as a flow
• Study the depth of decimation achieved
• WSAT as a measure of formula complexity
• Depends on details of tricks employed
• Shows SP produces renormalization out of hard-SAT
• With todays codes, WSAT outperforms SP!
• Except in regime where 1-RSB is stable
• WSAT has an endpoint at 4.15