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

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(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
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
Rho = 1
0 < Rho < 1
1 < Rho
BP
SP
BP  SP
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”
BP, SP, hybrids differ in their
“depth of decimation”
These results
are for SP
only
Depth of decimation achieved by BP,
hybrids…
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
Cost per spin is well-defined (linear)
WSAT cost/spin variance shrinks with N
Examination of
distributions
shows that
cost/spin is
concentrated as N
 infty up to
alpha = 4.15!
Cumulative distribution of cost/spin
alpha = 3.9
Cumulative distribution of cost/spin
alpha = 4.1
Cumulative distribution of cost/spin
alpha = 4.15
Cumulative distribution of cost/spin
alpha = 4.18
SP, like rule-based decimation, has an end-point
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