Going after the -SAT Threshold

1. Going after the-SAT Threshold Konstantinos Panagiotou (with Amin Coja-Oghlan)

2. -SAT Formulas • Given: • boolean variables • a Boolean formula in -conjunctive normal form(-CNF) whereis a variable orthenegationof a variable • An assignmentiscalledsatisfying(for), ifitsatisfies all clauses • A clauseissatisfied (by) ifat least oneliteral in itissatisfied

3. Example () • Assignment is satisfying • Assignment is not

4. The -SAT Problem • Question: given, isthereanysatisfyingassignment? • This is a centralproblem in computerscience. • If, then it is easy: • issatisfiableiffno variable appearsbothnegatedand not negated • If, then there is a linear time algorithm [Aspvall, Plass & Tarjan (1979)] • If , then the problem is -complete [Cook & Levin (1971)]

5. Random Formulas • Setup: • variables • is a -CNF withclauses, whereeachclauseisdrawnuniformlyatrandomfromthesetof all possibleclauses • Wecallthedensityoftheformula

6. A Generative Procedure • Generate as follows: • for // Generate • for// Generatethliteral in • , where is uar (uniformly at random) from • Withprobability set (i.e. negate the occurrence of the variable) • All randomdecisionsareindependent • Particularly, thechoiceofthe variable andofits „sign“ aredistinctprocesses

7. ManyQuestions… • For whichdensities (# clauses) is satisfiable whp (with high probability)? • Initial motivationforstudyingrandom-SAT: the „mostdifficult“ instancesseemtobearound a specific • Other propertiesthat hold whp? • Algorithms? • We will consideronlythecase here.

8. Picture - Satisfiability Pr[issatisfiable] as 1 ? 0 c (density)

9. A First Bound • Consider theobviousrandom variable = # ofsatisfyingassignmentsof • Ifforthefixedvalueofwecanshow as, thenand is not satisfiablewhp. • Let, wherethesumisover all possibleassignments in and

10. A First Bound (cont.) In

11. Picture Pr[issatisfiable] as 1 ? 0 c (density)

12. (Some) Previous Work • Friedgut ’05: Thereis a sharp thresholdsequence: • If, thenissatisfiablewhp • If, thenitis not whp • Kirousis et al. ’98: • Achlioptasand Peres ’04:

13. Beforeour Work Pr[issatisfiable] as 1 Gap: c (density) 0

14. OurResult Coja-Oghlan, P. ‘13: 1 Gap: 0

15. Random CSP‘s • Manyexamples • Variationsofrandom-SAT (NAESAT, XORSAT, …) • -coloringrandomgraphs • 2-coloring random-uniform hypergraphs • Fornorandomversionoftheseproblems(in theNP-hardcases) thethresholdisknown • Statistical physicistshavedevelopedsophisticated but non-rigoroustechniques • detailedpictureaboutthestructuralproperties • severalconjectures, algorithms • manypapers: Krzakala, Montanari, Parisi, Ricci-Tersenghi, Semerjian, Zdeborova, Zecchina, … • Mathematicaltreatment: Talagrand

16. OneConjecturefor-SAT Pr[issatisfiable] as 1 Gap: c (density) 0

17. The Second Moment Problem • If is a non-negative random variable • Wecanapplythisto, thenumberofsatisfyingassignmentsof • If for the given , then we are done! • Problem: forallwehavethatis exponentially larger than ! Paley-Zygmund Inequality Second Moment Method

18. Boundfor 2nd Moment

19. Why?

20. An Asymmetry • Consider a thoughtexperiment • Supposethatsomebodymakesthepromise „appears in exactly times … … andalltheseappearancesarepositive“ • Whatvalue do weassignto? • Other promise: „appears in exactly times … … and51%oftheappearancesarepositive“ • We (should) setagainto

21. The Majority • Our „bestguess“ for a satisfyingassignmentisthemajorityvote: • Somebodytellsushowofteneach variable appearspositivelyandnegatively, andnothingelse • If appears more often positively, assign it to , and otherwise to • This assignmentmaximizestheprobabilitythatissatisfied • Even more: assignmentsthatare „close“ tothemajorityvotehave a larger probabilityofbeingsatisfying

22. Picture ofthe Situation • Majorityassignment • Largestprobabilityofbeingsatisfiable • Distance 1 • Lessprobabilityofbeingsatisfiable • Distance 2 • Even smallerprobabilityofbeingsatisfiable

23. Symmetry vs. Asymmetry • Thereis an asymmetryin themeaningoftheassignment „true“ and „false“ • In manyotherproblemsthisisn‘t so. • In graphcoloring all colorsplaythe same role • In not-all-equal SAT therolesoftrueandfalsecanbeinterchanged • …

24. Getting a Grip on theMajority • Generate in twostepsasfollows: • Foreach variable chooserandomlythenumberofpositive occurencesandthenumberofnegativeoccurences. • Chooserandomly a formulawhereeach variable appears times positively and timesnegatively. • Want: distributions of arethe same. • Step 1 • Itis easy tosee in thatand are distributed like Po, and they are almost independent

25. Step 2 • How do wechoose a formulawhereeach variable appears times positively and timesnegatively? • Configurationmodel: Random Matching: variable occurencestopositions in clauses

26. A WeightingScheme • Let • measureshow „distinct“ themajorityvoteis • Lemma. . Proof. is a sumof (almost) independentrandom variables. • Lemma. Let . Then Proof. The numberofsatisfied variable occurences in themajorityassignmentincreaseslinearlywithw. • Conclusion: beatsthemaincontributiontoisfromhighlyatypical.

27. Ingredient Nr. 1 • This isthefirstingredient in ourproof: fix w! • Actually, we fix thewholesequence such that it enjoys many typical properties of independent Po random variables. • Generate in twostepsasfollows: • Foreach variable chooserandomlythenumberof positive occurencesandthenumberof negative occurences. • Chooserandomly a formulawhereeach variable appears times positively and timesnegatively. instead

28. More Things goWrong… • From now on letbethenumberofsatisfyingassignmentsof, where the degree sequence is typical. • Itturns out: secondmomentfailsagain. • Other parametersstarttofluctuate • Numberofunsatisfiedclausesunderthemajorityassignment • … • Need tocontroleverythingatonce.

29. Recall the Situation • Majorityassignment • Largestprobabilityofbeingsatisfiable • Distance 1 • Lessprobabilityofbeingsatisfiable • Distance 2 • Even smallerprobabilityofbeingsatisfiable

30. Our Variable – 2nd Ingredient • We do not countallsatisfyingassignments! • Intuition: if a variable appears times positivelyand times negatively, thenassignittotruewithsomeprobability that depends on only. • Map • Set also , • Meaning: a -fractionoftheliteralsissatisfiedundertheassignmentsthatweconsider.

31. More formally • Set • This isthesetof different „types“ of variable occurences (equivalent) • Wesaythat has -marginalsif for all • Thatis, a t-fractionofthe variable occurencesissettotrue, for all • Question: how do wechoose?

32. Pictorially

33. Detour: Physics • For letbethefractionofsatisfyingassignments in whichissettotrue in • ItisNP-hardtocompute • Accordingtophysicists: canbecomputedby a messagepassingalgorithmcalledBelief Propagation [Montanari et al ‘07] • So-calledReplicaSymmetric Phase: uniquefixedpointsolutionexists • Condition: density

34. Conjecture • Belief Propagation leads to a stronger prediction • Conjecture for up to an error of as • This stronger conjecture is not explicit form • it does depend on many parameters

35. Our Choice • This matchestheconjecture on the „bulk“ ofthe variables • Recall that • Exceptof a verysmallfraction, all other variables havetheproperty

36. Summary & Outlook • Wecandeterminethe replica symmetric -SAT thresholdwith high accuracy • We manage forthefirst time toget a grip on an asymmetricproblem • On thewayweusealgorithmicinsightsgainedbyphysicists • Catching the -SAT threshold?

37. Thank you! +