1 Message Passing and Local Heuristics as Decimation Strategies for Satisfiability Lukas Kroc, Ashish Sabharwal, Bart Selman (presented by Sebastian Brand) Symposium on Applied Computing March 2009

2 Combinatorial Search Procedures Search procedures for combinatorial problems such as SAT usually 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

3 A Study of Decimation-Based Satisfiability 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 me asurable properties that highlight differences. – evolution of problem 'hardness' during decimation – generation (or not) of unit clauses

4 Talk Outline The decimation process (for solving SAT) Decimation strategies  Local heuristics  “Global” message passing heuristics Empirical comparison Differences in decimation strategies

5 The Decimation Procedure Given some ordering of the variable-value pairs Do: 1. Assign the first variable its value 2. Simplify the problem instance 3. 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?

6 Random k-SAT Problem k-SAT Problem: Satisfiability of Boolean formula in CNF  n vars  {0,1}  {F,T}, m clauses, each clause has k literals  Is F satisfiable? Find a solution. Canonical NP-complete problem Random problem: clauses chosen at random  Phase transition, as a function of clause-to-variable ratio  For 3-SAT,  thr  4.26

7 Local Counting Heuristic Used in standard systematic solvers (DPLL) for SAT Variants of: “set the most frequently occurring variable based on majority vote” Fast, easy to compute, but not very powerful Local UNSATSAT

8 Global Statistical Heuristic Compute marginal probabilities: how often is x True in solutions? Pr[ x=True | a solution ]  Use magnetization as a heuristic: Pr[x=T|solution] - Pr[x=F|solution] Inference in a Bayesian Network:  More difficult than SAT! xyz ABC

9 Local UNSATSAT BP SAT as an Inference Problem Marginals can be approximated using Belief Propagation:  Message Passing: numbers for each (clause,var) pair governed by a system of equations.  Scalable, but does not work for hard enough problems due to loops in the factor graph Both convergence and accuracy issues Modified BP Want to close the gap!

10 Belief Propagation for Inference The original BP does not converge – first need to dampen it to force convergence Damping constant 1: same as BP 0: guaranteed convergence

11 Belief Propagation How well does it do “in general”?  Speed: either fast, or doesn’t converge  Accuracy: not very good (for hard problems)  Benchmark: large (#vars=5K) random 3SAT,  =4.2  Comparing “true” marginals vs. BP-generated marginals (Ideal case: straight diagonal line) Standard BP: Forced BP: Solution Marginals BP Marginals Solution Marginals BP Marginals Doesn’t seem very useful for SAT!

12 Using Surveys Instead of Beliefs BP-inspired-decimation does not work for very hard random instances SAT instance Solve it by decimation Use BP for Pr[x=T|solution] For more difficult random SAT problems, use SP-inspired-decimation Modify the problem itself SAT instance Solve it by decimation Use SP for Pr[x=T|”cover”]

13 Survey Propagation Much More Accurate! 5,000 variables (  =4.2) Marginals over “covers” computed much more accurately “covers” marginals very tightly coupled with solution marginals – good enough for our purposes!

14 First experimental study: How well do various decimation heuristics perform?

15 Results: 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.

16 Second experimental study: What are some measurable properties that provide insights 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

17 Generation of Unit Clauses Unlike all other heuristics considered, SP generates nearly no unit propagations until around 40% of the variables are set!

18 Generation of Unit Clauses Proposition: If 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.

19 Evolution 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!

20 Summary 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