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”

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/spinalpha = 4.1

Cumulative distribution of cost/spinalpha = 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