comparing beliefs, surveys, and random walks for 3-sat scott kirkpatrick, hebrew university joint...
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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)
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 today’s codes, WSAT outperforms SP! Except in regime where 1-RSB is stable
WSAT has an endpoint at 4.15
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
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(ia) = fraction of solutions s.t. variable
i satisfies clause a (influence) I(ai) = fraction of solutions s.t. clause a
is satisfied by variables other than I
I(ai) T(ja) + T(ka) – T(ja)T(ka) Same iteration for BP, SP
Closing the loop introduces a one parameter family of belief schemes
Calculate new T’s from the I’s, 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 < Rho < 1 BP SP 1 < Rho SP unknown
Interpret effects of Rho in flow diagram:
Visualize decimation as flows in the SP space
Decimate variables closest to the corners
Origin is the “paramagnetic phase”
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 < alpha < 4.267
We use WSAT (from H. Kautz, B. Selman, B. Cohen) as a standard measure of complexity
Results of SP + decimation:
Upper curves:
WSAT cost/spin
Lower curves:
WSAT cost/spin after decimation
(two normalizations)
Where does this pay off?
Using today’s 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
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
WSAT cost/spin variance shrinks with N
Examination of distributions shows that cost/spin is concentrated as N infty up to alpha = 4.15!
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 today’s codes, WSAT outperforms SP! Except in regime where 1-RSB is stable
WSAT has an endpoint at 4.15