Planted Random 3SAT with a Small Fraction of 1-Clauses

Transcription

1 Departmet of Computer Siee ad Applied Mathematis Weizma Istitute of Siee Plated Radom 3SAT with a Small Fratio of 1-Clauses Submitted for the degree of Master of Siee to the Sietifi Couil of the Weizma Istitute of Siee Uder the supervisio of Professor Uriel Feige Alia Arbitma February 2012

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3 Abstrat A plated radom 3SAT istae is formed by seletig a truth assigmet ad iludig eah lause osistet with it with a ertai probability. Whe the lause to variable ratio is Ω(log ) it is well kow that the assigmet a be reostruted by the Majority Vote heuristi, hee the more iterestig ase is whe the lause to variable ratio is ostat. I a paper from 2003, Flaxma preseted a modified versio of the plated model where lauses satisfied by differet umber of literals are iluded with differet probabilities. We fous o the ase where the umber of lauses satisfied by exatly oe literal is small, both i absolute ad i relative terms. We preset polyomial time algorithms for the two distributio families, where for the first oe we are also able to hadle a semi radom model for hoosig the polarities of the formula.

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5 Akowledgmets First ad foremost, I would like to express my deep gratitude to my advisor, Uriel Feige, for outstadig guidae ad eouragemet, for itroduig me with his luid way of thikig ad for a great sese of humour. May thaks to the Mathematis ad Computer Siee faulty, ad i partiular to Oded Goldreih for erihig ourses ad readig material ad Itai Bejamii for ispirig ourses ad oversatios. Thaks to my fellow studets at the faulty, for makig eah day at Weizma a true ejoymet. Last but ot least, I would like to thak my family ad my dear Maor, for their edless love ad support.

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7 1 Itrodutio The lassi problem of 3SAT is oered with fidig a satisfyig assigmet to a iput 3CNF formula i polyomial time. A 3CNF formula over the Boolea variables x 1,..., x is the ojutio of m lauses 1,..., m, where eah lause is the disjutio of 3 literals, i = l i,1 l i,2 l i,3, ad eah literal l i,j is either a variable or its egatio (we deote the egatio of x by x). A 3CNF formula is satisfiable if there is a assigmet of variables to {T rue, F alse} so that every lause otais at least oe literal assiged T rue. The 3SAT problem is well kow to be N P-Complete ad o algorithm a sueed o all 3SAT istaes i polyomial time, uless P = N P ([11],[22]). the worst ase has lead to a extesive average ase researh. The itratability of 3SAT i We oetrate o a probabilisti model for geeratig radom 3CNF istaes amed plated radom 3SAT, ad preset algorithms for a previously suggested variat of that model. A plated radom istae of 3SAT is formed by seletig a truth assigmet φ o variables uiformly at radom, ad the iludig eah lause satisfied by the plated assigmet φ with probability p. I a paper from 2003 Flaxma exteded this model, suggestig to assig differet probabilities aordig to the umber of literals i the lause that are satisfied by φ ([16]). Let p i be the probability to ilude a lause with exatly i literals satisfied by φ. Flaxma showed that for ay ostats η 2, η 3 [0, 1] there is a ostat d mi suh that for all d > d mi his spetral algorithm fids a satisfyig assigmet whp over istaes with p 1 = d 2, p 2 = η 2d ad p 2 3 = η 3d. The algorithm relies o there beig some positive fratio 2 of lauses with exatly oe satisfied literal ad does ot hadle the ase where p 1 < mi{p 2, p 3 }. We exted Flaxma s result by addressig two separate rages i whih p 1 < mi{p 2, p 2 }: 1. p 2 + p 3 = d 2 ad p 1 large d. for a uiversal ostat ad suffiietly (d log d)2 For somewhat smaller values of p 1 we are also able to deal with a semi-radom versio of the model. 7

8 2. p 1 = d 1 ad p p 3 = d 2 with 2d 2 1 d 2 2 d 1 for a uiversal ostat ad suffiietly large d 1. The leadig ostat i 2d 1 d 2 is to some extat arbitrary 1. Next we defie the exat models i use ad state the mai result. 1.1 The Model Defiitio 1 (i-lause). For i {0, 1, 2, 3} we say a lause is a i-lause w.r.t assigmet ψ (to the variables that appear i ) if the umber of literals i satisfied by ψ is i. For i {1, 2, 3} we may also say that ψ satisfies as a i-lause. We osider two models for geeratig 3SAT istaes: Defiitio 2 (Plated Radom 3SAT Model). I this model a istae of 3SAT is formed by first hoosig a truth assigmet o variables uiformly at radom, ad the seletig eah i-lause for i {1, 2, 3} idepedetly with probability p i, for some p i [0, 1]. Defiitio 3 (Plated Semi-Radom 3SAT Model). I this model a istae of 3SAT is formed by first hoosig a truth assigmet o variables uiformly at radom ad the seletig m lauses as follows: for eah lause its variables are hose idepedetly at radom; the their polarities may be hose adversarially, as log as all lauses are satisfied ad the umber of 1-lauses does ot exeed ɛm. Note: There are two seletio proesses uderlyig the models desribed above: i oe, every i-lause is seleted idepedetly with a appropriate probability; i the other, m lauses are seleted idepedetly from all legal triples of literals (with a appropriate fratio of i-lauses for every i). We shall use the two 1 For most values of p 1, p 2 ad p 3 for whih d 2 2d 1 the aalysis of Flaxma s algorithm applies, ad hee our work does ot address this rage of parameters. There is a rage of parameters with d 2 2d 1 where either Flaxma s algorithm or ours work, amely, whe p 1 = p 3 ad p 2 is small. For suh values the bottleek is i fidig a approximate assigmet (the first stage of the algorithm). 8

9 proesses iterhageably throughout the work, for the sake of simpliity of the presetatio i questio. 1.2 Our Result We shall later itrodue three algorithms: Alg1, Alg2 ad Alg3; regardig these algorithms we prove the followig results. Theorem 1. There exists a ostat < 1 suh that for every suffiietly large d the followig holds: let F be a plated radom 3SAT formula geerated aordig to the radom model with p 1 (d log d) ad p 2 2 +p 3 = d, the the Alg1 algorithm 2 fids a satisfyig assigmet to F i polyomial time whp over the hoie of F. Theorem 2. There exists a ostat < 1 suh that for every suffiietly large d the followig holds: let F be a plated radom 3SAT formula geerated aordig to the semi-radom model with m = d ad ɛ d 4 log 2, the the Alg2 algorithm d fids a satisfyig assigmet to F i polyomial time whp over the hoie of F. Remark: Theorem 2 still holds eve if F otais also 0-lauses (w.r.t the plated assigmet), as log as the umber of 0- ad 1- lauses together does ot exeed ɛm. Theorem 3. There exists a ostat < 1 suh that for every suffiietly large d 1 ad for d 2 with 2d 1 d 2 2 d 1 the followig holds: let F be a plated radom 3SAT formula geerated aordig to the radom model with p 1 = d 1 2 ad p 2+p 3 = d 2, the the Alg3 algorithm fids a satisfyig assigmet to F i polyomial time 2 whp over the hoie of F. 1.3 Related Work ad Motivatio I the Radom 3SAT problem a 3CNF formula o Boolea variables is geerated by seletig m radom lauses idepedetly ad uiformly from all triples of literals. The goal is to fid a assigmet of variables to truth values so that the etire formula is satisfied, or prove that o suh assigmet exists. The probability 9

10 of a istae draw from this distributio to be satisfiable has a iterestig oetio to m/, whih is the lause to variable ratio (ofte also referred as the lause desity ). Whe the ratio is low, the istaes are likely to be satisfiable, whereas whe the ratio is high, the istaes are usatisfiable with high probability. This satisfiability threshold is kow to lie betwee 3.42 [19] ad 4.5 [20] ad experimets suggest that it is approximately 4.2 [12]. This is ot yet prove, but Friedgut [18] has show that there exists a sequee γ suh that if m/ γ ɛ the probability of a istae (with suh desity) to be satisfiable teds to 1 ad if m/ γ + ɛ this probability teds to 0. It is still ot kow, however, if the sequee γ overges (or, equivaletly, a be take as a ostat threshold). Several algorithms are kow to perform well for istaes draw from the radom 3SAT distributio with low lause desity. Oe suh algorithm is the Pure Literals Heuristi, for whih a desity as low as d 1.63 suffies ([23]). Aother greedy algorithm sueeds with asymptotially positive probability for d 3.42 ([19]). Furthermore, experimetal results suggest that algorithms from the Survey Propagatio family sueed for desity very lose to the ojetured satisfiability threshold ([8]). Whe evaluatig algorithms, we geerally require they perform well o most istaes draw from a ertai distributio. For istaes with desity above the threshold, this meas we expet them to fid satisfyig assigmets for a small fratio of the probability spae, as oly suh fratio of istaes is satisfiable. It is thus atural to osider the oditioal distributio, where a 3SAT istae is formed by first seletig a formula at radom ad the keepig it oly if it is satisfiable. But ufortuately this distributio is both diffiult to sample from ad to aalyse. This has lead to researh o plated radom 3SAT. I the plated model first a truth assigmet o variables is seleted uiformly at radom (this assigmet is referred to as the plated assigmet ) ad the the formula is seleted at radom from all formulas whih are satisfied by this assigmet. Oe seletio proess whih guaratees the resultig formula is ideed satisfied by the plated assigmet is pikig m lauses idepedetly ad uiformly at radom oly from all triples of literals whih are satisfied by the plated assigmet. I geeral, for a give triple of variables, there are 2 3 = 8 ways of hoosig the variables polarities (for every variable x we take either x or its egatio x). If 10

11 we are ostraied by a assigmet hose beforehad, the this umber redues to 7, sie there is exatly oe possible hoie of polarities for whih all literals are ot satisfied by the assigmet (resultig i the etire lause beig usatisfied), ad that hoie should ot be iluded i the istae. Aother atural seletio proess is hoosig every lause satisfied by the assigmet idepedetly with a ertai probability p (sometimes it may be more oveiet to thik of this proess as proeedig i two stages: o the first stage we pik three variables at radom ad the selet their polarities out of the 7 legal possibilities). Whe omparig the plated distributio to the uiform distributio o satisfiable istaes, we otie that the umber of assigmets a istae has ireases its probability to be seleted i the plated model, whereas i the uiform distributio learly all satisfiable istaes are seleted with the same probability. For a lause desity as high as Ω(log ) a radom satisfiable istae is likely to have oly oe satisfyig assigmet, whih may explai why i that rage the two distributios are statistially lose ([5]). A justifiatio to ivestigate the plated distributio for a ostat desity may be foud i a reet work, showig that for suh desity (with suffiietly large ostat) the two distributios possess a similar struture of solutio spae, implyig that i may ases algorithms for the plated distributio may be applied for the uiform oe as well ([10]). Plated distributios have bee the fous of researh i several differet otexts: plated graph olorig ([6],[3]), plated bisetio ([7]), plated lique ([14], [4]) ad plated 3SAT ([21],[16],[15]). Some of the works ispired ours, ad espeially those of Flaxma ([16]) ad Alo ad Kahale ([3]) whose tehiques are applied here. I the otext of 3SAT, for istaes draw from the plated distributio with lause desity as high as Ω(log ), the Majority Vote reostruts the plated assigmet whp ([21]). This heuristi assigs to eah variable the truth value that satisfies the majority of the lauses i whih it appears. The motivatio behid it the followig: if every lause osistet with the plated assigmet is iluded with the same probability, the there is a bias towards iludig the literal satisfied by the plated assigmet more frequetly tha its egatio. For lower desities (i.e., ostat oes) the Majority Vote would ot satisfy the etire formula but may serve as a good startig poit, ad the the k-opt heuristi a omplete 11

12 the assigmet, as demostrated i [15]. Whe the probabilities to ilude eah i-lause i the istae are set appropriately, the Majority Vote fails, but i suh ase spetral steps apply ([16]). As metioed, Flaxma suggested a geeralized versio of the plated model where lauses with differet umbers of satisfied literals are iluded with differet probabilities. A further appealig geeralizatio osiders semi-radom models, wherei the uderlyig priiple is i there beig a mixture of radom ad adversarial elemets, with a varyig proportio of the two. I geeral, the larger the portio of the adversarial elemets our algorithms a withstad, the larger the probability spae treated by them is, ad hee the more robust they are. Semi-radom models have bee suggested ad studied for several problems ivolvig plated distributios, ad amog others, idepedet sets ad graph bisetios ([13]) ad graph olorig ([6]). I the otext of 3SAT, Vilehik ad Feige [15] osidered a adversary who is allowed to add arbitrary 3-lauses to a previously geerated plated radom 3SAT istae with a ostat lause desity. I our work oe of the models addressed is a semi radom oe, i whih first the variables of eah lause are hose radomly ad the a adversary is allowed to hoose their polarities, as log as the fratio of 1-lauses is small. 2 Defiitios Defiitio 4 (Support). We say a variable x supports a lause w.r.t assigmet ψ if is a 1-lause w.r.t ψ ad x (or x) is the satisfyig literal of w.r.t ψ. Defiitio 5 (Partial assigmet). Let X = {x 1,..., x } be a set of Boolea variables. A partial assigmet to X is a elemet of {T rue, F alse, }, where we regard a variable assiged as uassiged. We say a lause is satisfied by a partial assigmet ψ (to the variables appearig i it) if it is satisfied by ay of its literals assiged i ψ. Defiitio 6 (-expasio). We say a set of k lauses has -expasio if the umber of distit variables i the set is at least k. I suh ase we may also say that the set is -expadig. 12

13 Defiitio 7 (Formula graph). We assoiate with a formula F the followig atural graph: the verties represet the variables ad two verties share a edge if there is a lause i F whih otais their variables. We all this graph the formula graph of F (i fat it is a multi-graph). 3 The Algorithms Similar to the approahes i [3], [16], our algorithms proeed i three mai steps: fidig a approximate assigmet, a Uassigmet phase ad ompletig the partial assigmet to oe that satisfies all lauses. For eah step we have several differet proedures possible. 3.1 Approximatio Majority Vote Give a 3CNF formula F, for every variable x ompute the Majority Vote for it, that is, out both the umber of positive ad egative ourrees of x i the formula. If the first outer is larger tha the seod oe assig T rue to x. Otherwise assig F alse. Redutio to MAX2SAT as follows: Give a 3CN F formula F, redue it to M AX2SAT 1. Covert F to a 2CN F formula by trasformig eah lause, say, (x y z), to (x y) (x z) (y z); 2. Apply o the 2CN F formula a approximatio algorithm for M AX2SAT kow to have the followig guaratee: if the iput formula is (1 ɛ)- satisfiable the the assigmet output by the algorithm will satisfy at least a 1 O( ɛ) fratio of all lauses. The take the assigmet retured by the algorithm applied. For example, the algorithm that was suggested by Charikar, Makaryhev ad Makaryhev i 2009 has suh guaratee ([9]). 13

14 3.2 Uassigmet 0-Clause Uassigmet Let ψ be the approximate assigmet from the previous step. While there are lauses usatisfied by ψ, 1. Form a partial assigmet ψ from ψ by uassigig all variables that appear i suh lauses. 2. ψ ψ. Cosider the fial partial assigmet (after the last iteratio), deote it by σ. The we otie there are oly two types of lauses w.r.t σ: satisfied lauses (i whih at least oe of the variables is assiged) ad uassiged lauses - lauses whose variables are all uassiged i σ. Small-Support Uassigmet from the previous step. Let agai ψ be the approximate assigmet 1. Form a partial assigmet ψ from ψ by uassigig all variables whih support less tha d 1 2 lauses w.r.t ψ. 2. While there are variables whih support less tha d 1 3 w.r.t ψ, fully assiged lauses (a) Form a partial assigmet ψ from ψ by uassigig all suh variables. (b) ψ ψ. 3.3 Completig the partial assigmet Mathig Let σ be the fial partial assigmet of the 0-Clause Uassigmet phase ad osider the set of uassiged lauses w.r.t σ. Costrut a bipartite graph for this set of lauses, as follows: o the left had side we will have oe vertex for eah variable ad o the right had side oe vertex for eah lause. A left had side vertex ad a right had side vertex share a edge oly if the orrespodig 14

15 variable appears i the orrespodig lause. Fid a maximum mathig i the graph. If every right had side vertex is mathed i this mathig, assig eah variable aordig to the demad of the lause mathed to it. Otherwise, fail. Note that a maximum mathig i a bipartite graph may be foud effiietly. Exhaustive Searh Let σ be the fial partial assigmet of the Small-Support Uassigmet phase. First simplify F as follows: set all variables assiged i σ aordig to their assigmet, remove all satisfied lauses ad remove the assiged variables from the remaiig lauses. If this results i a empty lause, fail. Otherwise, osider the simplified formula yet to be hadled. Notie it has three types of lauses: lauses with 1, 2 or 3 literals. Next perform a Uit Propagatio, that is, apply iteratively the followig: 1. If there is a lause with a sigle literal, set its variable as required by the polarity. 2. Simplify the formula as previously. 3. If this results i a empty lause, fail. At the ed of this proedure we are left oly with lauses of legth 2 or 3. Cosider the formula graph idued by the formula at had. If it otais a oeted ompoet of size larger tha log, fail. Otherwise look for a satisfyig assigmet by performig a exhaustive searh over the variables i eah oeted ompoet of this graph separately. If o assigmet satisfies the formula, fail. Otherwise retur the satisfyig assigmet. 3.4 The algorithms The three algorithms proeed as follows: Alg1 1. Majority Vote 2. 0-Clause Uassigmet 3. Mathig Alg2 1. Redutio to MAX2SAT 15

16 2. 0-Clause Uassigmet 3. Mathig Alg3 1. Majority Vote 2. Small-Support Uassigmet 3. Exhaustive Searh 3.5 Algorithms Overview Our work addresses two differet 3SAT distributio families: i oe, the absolute umber of 1-lauses is small (p 1 O(d log d) 1 2 ), while i the other this umber is small i relative terms (p 1 = d 1 2 for d 1 d 2 /2). For the first distributio, we osider both a radom ad a semi-radom model versios. For the radom model (i both distributio families), our algorithms begi by applyig the demorati proedure whih outs for eah variable both the umber of lauses i whih it appears as a positive literal ad those i whih it appears egatively (it treats eah suh ourree as a vote ) ad makes its deisio based o the majority preferee. A deliate hoie of the probabilities to ilude eah i-lause i the formula might fool the Majority Vote. That is the reaso why Flaxma s algorithm begis with spetral steps istead. I our ase, however, the fratio of 1-lauses is small eough for this heuristi to be appliable. I the semi-radom model a adversary hooses the polarities ad so she a tilt the statistis of a large liear set of variables, ausig their Majority Vote to be uiformative. We overome this obstale by exploitig the small umber of 1-lauses i a differet maer; we observe that the formula redued to its 2- ad 3-lauses is i fat a satisfiable 2SAT istae, ad hee a MAX2SAT approximatio algorithm may be applied o the etire formula. I suh ase the iitial umber of 1-lauses should be somewhat smaller tha i the radom model. A iterestig feature of the model versio i whih the fratio of 1-lauses is small i absolute terms is that all 1-lauses may be replaed by 0-lauses. I 16

17 suh ase the plated assigmet itself does ot satisfy the formula, but our proofs show that a satisfyig assigmet exists ad moreover a be foud i polyomial time. For a lause desity whih does ot deped o (as we osider), the Majority Vote is ot likely to satisfy the formula ad therefore a orretio is required; ideed i our algorithms it is followed by a uassigmet proedure. For the rage i whih the umber of 1-lauses is small i absolute terms (p 1 O(d log d) 1 2 ), we osider a very atural iterative proess whih we all 0-Clause Uassigmet : it begis by uassigig all variables that appear i lauses usatisfied by the Majority Vote ad does it iteratively util all lauses are satisfied by the obtaied partial assigmet. For a suffiietly small umber of iitial 1-lauses, the umber of lauses whose variables would be uassiged durig suh proess is small. This esures, for a istae whih was iitially geerated i a radom maer, that the residual formula possesses a partiular epesio property, whih guaratees the existee of a lause to variable mathig (by Hall s theorem). I suh ase we are able to omplete the assigmet by assigig eah variable aordig to the preferee of the lause mathed to it. Whe the umber of 1-lauses is small oly i relative terms (d 1 d 2 /2), however, the uassigmet proedure desribed above does ot have to result i a small umber of uassiged lauses ad therefore a differet kid of uassigmet is required. For suh ase we osider aother atural proedure, ispired by [ 16], whih we all Small-Support Uassigmet : at every iteratio all variables that do ot support eough fully assiged lauses are uassiged. This proedure is based o 1-lauses ad as suh it has the followig useful property: every variable whih is assiged by the Majority Vote differetly tha by the plated assigmet ad has also survived the uassigmet must support may lauses, eah of whih otais aother variable o whih the two assigmets disagree. The the subgraph idued by suh variables would have more edges tha expeted from a similar-sized subgraph i a radom formula. Ideed it is this property that guaratees o variables o whih the Majority Vote disagrees with the plated assigmet survive the uassigmet phase. To omplete the assigmet we osider the residual formula graph ad perform a very simple proedure: exhaustive searh over all oeted ompoets of this 17

18 graph. Whe the umber of uassiged variables at the ed of the previous step is small (as we are able to show), this graph does ot otai ay oeted ompoet larger tha log whp. Sie the uassigmet proedure is based o 1-lauses, whose umber depeds o d 1, whereas the oeted ompoets are idued by 2- ad 3-lauses as well (whose umber depeds o d 2 ), here we eed a additioal assumptio restritig the umber of 2- ad 3-lauses as a futio of the umber of 1-lauses: d 2 2 O(d1). It is ot lear to us how this assumptio a be removed. Ireasig the umber of 2- ad 3-lauses will improve the Majority Vote further, but might also result i larger tha log -sized ompoets i the residual formula graph. I suh ase a differet proedure tha a exhaustive searh may be eeded for ompletig the assigmet. 4 Corretess 4.1 Approximatio Lemma 4.1. Let F be a plated radom 3SAT formula geerated aordig to the radom model with p 1 = d 1, p p 3 = d 2, where the atual parameters are as i 2 Theorem 1 or 3. The whp over the hoie of F the Majority Vote disagrees with the plated assigmet o at most 2 Ω(d2) variables. Proof. Take ay variable x. First we show that the Majority Vote maj ad the plated assigmet φ disagree o its assigmet with probability 2 Ω(d2). Assume φ(x) = T rue ad fix two more variables y ad z. We are iterested i the umber of lauses osistig of the three variables x, y ad z, where x appears as a positive literal (x) ad i those where it appears egatively ( x). I total we have four suh lauses with positive ourrees of x: oe 1-lause, two 2-lauses ad oe 3-lause, ad three lauses with egative ourrees: two 1-lauses ad oe 2- lause. Let p x be a radom variable outig the atual umber of lauses i F where x appears positively ad similarly x for egative ourrees. The E[p x ] = (p 1 + 2p 2 + p 3 ) 2 ad E[ x ] = (2p 1 + p 2 ) 2. The Majority Vote disagrees with φ o the assigmet of x whe p x x. Sie E[p x ] E[ x ] = (p 2 +p 3 p 1 ) 2 = d 2 d 1 d 2 /2, where the last iequality is due to the assumed parameters i Theorem 18

19 1 or 3, ad both p x ad x are biomial radom variables, we have P r[p x (E[p x ] + E[ x ])/2] 2 Ω(d 2) ad similarly P r[ x (E[p x ] + E[ x ])/2] 2 Ω(d 2). We olude that P r[p x x ] 2 Ω(d 2). Next from liearity of expetatio, the expeted umber of variables o whih maj ad φ disagree is 2 Ω(d 2) ad by Markov iequality this happes with probability 1 2 Ω(d 2). A stroger oetratio result (with probability that depeds oly o ) may be obtaied by lookig at the proess of lauses seletio as a martigale with bouded differee ad applyig Azuma s iequality; osider the proess i whih m lauses are seleted idepedetly at radom. Let M i deote the umber of variables o whih maj ad φ disagree up to the seletio of the i-th lause, for i [1, m] (the M m is the total umber of variables o whih maj ad φ disagree). Sie eah ew seleted lause a effet the Majority Vote of at most 3 variables, it holds that M i M i+1 3 for all i. I suh) ase Azuma s iequality guaratees that P r[ M m E[M m ] t] 2exp ( t2. Pluggig i t = E[M m ] we obtai 18m that with probability 1 e Ω() 2 Ω(d 2 ), M m is ideed 2 Ω(d2) as expeted. Lemma 4.2. Let F be a plated radom 3SAT istae as i Theorem 1. The whp over the hoie of F the umber 0- ad 1-lauses w.r.t the Majority Vote is at most (ad the rest are 2- or 3-lauses). O(d log d) Proof. I the followig aalysis we osider the model i whih m = d lauses are piked uiformly at radom. The 1-lauses of F (w.r.t the plated assigmet φ) may serve as 0- or 1-lauses w.r.t the Majority Vote maj as well, ad their umber is. Apart from O(d log d) these lauses, ay 2- or 3-lause w.r.t φ that maj ad φ disagree o oe or more of its variables might also otribute to the out of 1- ad 0-lauses w.r.t maj. Hee we are iterested i boudig the umber of suh lauses. Look at all variables upo whih maj ad φ disagree. We would oditio o the evet there are 2 αd suh variables for some ostat α, as guarateed by Lemma 4.1. Let X be a fixed set of 2 αd variables. The average degree of X may be expressed by 1 X x X, F 1 {x } (where the summatio is over all lauses of F ad variables of X ad 1 {x } represets the idiator variable of the evet x appears i the lause 19

20 ). [ 1 E X x X, F 1 {x } ] = 1 X x X, F P r[x ] = 1 2 αd 2 αd d 3 = 3d. Sie the seletio proess of the lauses is idepedet, by Cheroff boud we kow that for δ > α/2 with probability 1 2 Ω(αd2 αd ) the average degree of X is at most (1 + δ)3d. Takig the uio boud over all sets of size 2 αd, we obtai the probability that ay suh set has a average degree larger tha (1 + δ)3d is upper bouded by the followig expressio: ( ) ( e ) 2 αd 2 Ω(αd2 αd ) 2 Ω(αd2 αd ) 2 2 O(d) 2 αd 2 αd We olude that whp the umber of lauses i whih all disagreed variables appear is at most O(d)2 Ω(d) = 2 Ω(d). Lemma 4.3. Let F be a plated semi-radom 3SAT istae as i Theorem 2. The whp over the hoie of F the umber of 0- ad 1-lauses w.r.t the approximate assigmet foud by Alg2 i the first step (Redutio to MAX2SAT) is at most (ad the rest are 2- or 3-lauses). O(d log d) Proof. Reall that to obtai a approximate assigmet for F, Alg2 first trasforms it to a 2SAT form ad the applies a kow tehique for approximatig MAX2SAT. Take ay lause, say, (x y z), ad assume it is satisfied by the plated assigmet as a 2- or 3-lause. The for ay two literals of this lause, say, x ad y, the plated assigmet would also satisfy their disjutio (x y), ad hee it must satisfy the followig 2SAT form as well: (x y) (x z) (y z). Thus if we trasform eah lause to a 2SAT form i suh maer, we may ow view the etire formula as a MAX2SAT havig a assigmet that satisfies a 1 ɛ fratio of the lauses for ɛ = d 4 log 2 (sie we are guarateed that i the d origial formula oly a ɛ fratio of the lauses are 0- or 1-lauses w.r.t φ, ad the rest are 2- or 3-lauses). The approximatio algorithm for MAX2SAT used i the Redutio to MAX2SAT phase of Alg2 satisfies a 1 O( ɛ) fratio of all lauses 20

21 ([9]). I our ase applyig suh algorithm would result i a assigmet w.r.t whih the umber of 0- ad 1-lauses is at most O( ɛ)m = O( d 4 log 2 d)d = O(d log d). 4.2 Some tehial lemmas Lemma 4.4. There exists a ostat suh that for every suffiietly large d the followig holds: let F be a 3SAT formula with d lauses ad assume the variables of eah lause were hose idepedetly at radom. The whp over the hoie of F every subset of i lauses for i otais at least i distit variables. d Proof. We would like to determie the largest possible m for whih whp over the hoie of F every set of i lauses for i m otais at least i distit variables. We would boud the omplemet by osiderig a evet with eve greater probability, that is, the existee of a set of suh size whih otais at most i distit variables. Formally we ask whe does the followig expressio overge to 0 as teds to : S = m ( d i i=4 )( 3i i ) ( i For every i we hoose the i lauses out of the possible d, fix i variable positios (these are the distit adidates) ad require variables i all other positios are hose out of these adidates. We start the summatio from 4 sie every set of 1, 2 or 3 lauses otais at least 3 distit variables (o repeated variables withi a lause). Note that here we assume the simpler to aalyse model, i whih for every lause its variables are hose idepedetly oe of aother (this proess will result whp i at most O(d) ivalid lauses with two idetial variables eah, whih would be exluded from the formula). We upper boud this expressio usig ( ) k ( e k )k to obtai: ) 2i S m a i = i=4 m ( ) i ( ) i ( ) 2i ed 3ei i = i i i=4 21 m ( i (3e 2 d) i i=4 ) i

22 We otie that a i+1 a i = ) i+1 ( (3e 2 d) i+1 i + 1 ( ) i = 3e 2 d i (3e 2 d) i ( i ) i ( ) i + 1 ( ) i + 1 3e 2 d e 1. Takig m = with a appropriately hose leadig ostat i O(d) (6e3 O(d) should be eough), we guaratee that a i+1 q for some q < 1. a i Hee this sum may be bouded by a ifiite geometri oe, as follows: S a 4 1 q = (3e2 d) 4 ( 4 )4 1 q = (12e2 d) 4 1 q ( ) 1 1 = O = o(1). 4 poly() We olude that every subset of i lauses for i distit variables whp. O(d) otais at least i Lemma 4.5. There exists a ostat > 0 ( = 12 suffies) for whih the followig holds: let F be a plated radom 3SAT istae as i Theorem 3 ad osider the formula graph assoiated with F. The whp over the hoie of F every vertex idued subgraph of size as small as has a average degree of at most. O(d 2 ) Proof. Cosider a set of variables S whih idues a subgraph with average degree of at least = 6, the this subgraph must otai at least 3 S edges ad hee there are k lauses otaiig at least two variables from S eah, for k [ S, 3 S ] (eah lause orrespods to either 1 or 3 edges of the graph). The total umber of lauses otaiig at least two variables from S whih are also satisfied by the plated assigmet is l = 7 ( ) S 2, so for every k the probability at least k suh lauses are atually iluded i F is at most ( l k) (max{p1, p 2, p 3 }) k ) (d2 2 ) k. We take the uio boud over all possible values of k ad over all ( l k sets S of size up to O(d 2 2 ) to estimate the probability of ay suh set to exist. S = O(d 2 2 ) i=6 ( ) 3 i b k = i k= i O(d 2 2 ) i=6 22 ( ) 3 i i k= i ( ( 7 i ) ( 2) d2 k 2 ) k

23 3 i k= i b k = 3 i k= i 3 i k= i 3 i k= i 3 i ( ( 7 i ) ( 2) d2 k 2 ) k ( ) 7ei 2 k ( ) k d2 2k 2 ( ) 7ei 2 k ( ) k d2 = 2 i 2 ( ) k 7ed2 i = 2 k= i ( ) k 7ed2 i = 2 k= i ( 7ed2 i = 2 ( 7ed2 i 2 ( ) 7ed2 i i = 2 2 ) i ( 1 7ed 2 2 ) i ( 1 1 2) 1 = ) 1 i where the last iequality is justified by i O(d 2 2 ) with appropriately hose ostat. Thus we obtai: S 2 O(d 2 2 ) i=6 a i = 2 = 2 O(d 2 2 ) i=6 O(d 2 2 ) i=6 ( e i ( e +1 ) i ( 7ed2 2 ( 7d2 2 ) i i ) ( ) ) i 1 i 23

24 Next we would boud this sum by a ifiite geometri oe. a i+1 a i = e +1 ) ( 1)(i+1) ( i + 1 ( ) 7d2 2 ( ) ( = i 1)(i) ) i( 1) ( ) i ( ) i = ( ) i ( = O(d 2 ) i O(d 2 )e 1 = O(d 2 ) For 2 (whe the average degree = 6 is at least 12) ad by hoosig the appropriate ostat i i O(d 2 2 ) (a ostat of e 4 would be suffiiet) we may boud O(d 2 ) ( ) i+1 1 by some q < 1, implyig that S 2a 6 1 q = poly() = o(1) Lemma 4.6. There exists a ostat > 0 for whih the followig holds: let F be a plated radom 3SAT istae as i Theorem 3 ad osider the formula graph assoiated with F. The whp over the hoie of F every vertex idued subgraph of size as small as, where oly edges orrespodig to 1-lauses w.r.t the O(d 1 ) plated assigmet are osidered, has a average degree of at most. Proof. The proof is idetial to that of Lemma 4.5 but osiders sets of at most O(d 1 1 ) variables ad replaes max{p 1, p 2, p 3 } by p 1 = d Uassigmet Lemma 4.7. Let F be a 3SAT formula with d lauses ad assume the variables of eah lause were hose idepedetly at radom. Cosider a arbitrary assigmet ψ w.r.t whih the umber of 0- ad 1-lauses is at most O(d log d). The 24

25 whp over the hoie of F a 0-Clause Uassigmet whih begis with ψ results i at most uassiged lauses. O(d) Proof. Let σ be the fial partial assigmet of the 0-Clause Uassigmet phase (it is partial to ψ, the iitial assigmet of that phase). I the followig aalysis a i-lause is suh w.r.t the iitial assigmet ψ whereas a uassiged lause or variable is suh w.r.t the fial partial assigmet σ. Let m iit be the umber of 0- ad 1-lauses, ad m ed be the umber of uassiged lauses at the ed of the uassigmet. Our objetive is to upper boud m ed by a futio of m iit. Let m ed = m 01 + m 23, where m 01 is the umber of uassiged lauses whih are 0- or 1-lauses (w.r.t ψ) ad similarly m 23 is the umber of uassiged lauses whih are 2- or 3-lauses (w.r.t ψ). The umber of distit variables that appear i the uassiged lauses is at most 3m 01 +m 23, sie eah 0- or 1-lause otributes at most three distit variables, whereas eah 2- or 3-lause, i the momet it beomes usatisfied (ad as a result uassiged), must otai at least two variables already uassiged (otherwise it ould ot beome usatisfied), thus otributig at most oe ew variable. We assume towards a otraditio that m ed m 01 for some ostat. For the sake of aalysis, let us termiate the proess at the very iteratio whe m ed = m 01. Also assume for simpliity all 1-lauses are uassiged, that is, m 01 = m iit. Pluggig i the parameters aordig to our assumptio we obtai that the umber of distit variables is at most 3m 01 +m 23 = 3m 01 +(m ed m 01 ) = 3m 01 + (m 01 m 01 ) = ( + 2)m 01 ; i other words, the maximal possible expasio of the set of uassiged lauses is Our strategy is to show that for small eough m iit every set of size at most m iit is whp at least (1 + 2)-expadig, whih would imply that m ed must be i fat smaller tha m iit. To be more orete, we are lookig for the largest m ed suh that every set of at most m ed lauses is at least (1 + 2 )-expadig whp. Similarly to Lemma 4.4, we ask what is the maximal m ed for whih the followig sum overges to 0 whe teds to : S = m ed i=3 ( )( d 3i ) ( (1 + 2 i (1 + 2)i ) [3 (1+ 2 )]i For every i we hoose the i lauses out of the possible d, fix (1 + 2 )i variable 25

26 positios, whih serve as the distit adidates, ad require all variables i other positios are hose out of these adidates. Next, we upper boud this expressio usig ( ) k ( e k )k to obtai: S = m ed i=3 m ed i=3 a i = ( m ed i=3 ( ) i ( ed 3ei ) (1+ 2 )i ( (1 + 2 i (1 + 2)i ( ) 1 4 ) i ( ) (1 2 e 2+ 2 i )i d ; ) (2 2 )i = Hee, a i+1 a i = = = ( ( ( ( ( ) ( ( i+1 ( i + 1 e 2+ 2 d) ) i ( (1 2 i ) ( ) e 2+ 2 d ) 1 4 e 2+ 2 d ) ( i ( ) ) 1 4 e 2+ 2 d e 1 2 ( ) 1 4 e 3 d) (i + 1 ) (1 2 )(i+1) ) i(1 2 ( i + 1 ) 1 2. )i = ) ( i + 1 ) 1 2 = ) 1 2 This time we eed to hoose m ed < ( ( ) e 3 d ) 2 to have a i+1 a i q 26

27 for some q < 1, so we a boud S by the followig ifiite geometri sum: S a 3 1 q = ( 3 ( ( = 1 q ( ) 1 = O poly() = o(1). ( ) ) e 2+ 2 d ) 1 4 e 2+ 2 d 1 q ) 3 1 (1 2 )3 = ( ) (1 2 3 )3 = To omplete the proof, we require that m ed m iit O(d). Cosiderig the maximal m ed possible, that is,, we obtai the ostrait: O(d 2 ) 1 d 2 d, or, d 2 2 whih implies ( ) log d Takig = O log log d algebrai maipulatios). We olude that whe m iit ( 2) log 2 log d. we guaratee the above ostrait is met (by simple O(d log d) the m ed O(d). Lemma 4.8. Let F be a plated radom 3SAT istae as i Theorem 3. The whp over the hoie of F, at the ed of the Small-Support Uassigmet phase of Alg3 there are at least (1 2 Ω(d 1) ) assiged variables ad the assigmet of all assiged variables agrees with the plated assigmet φ. Proof. The proof osists of two parts. First we idetify a set of variables of size (1 2 Ω(d 1) ) o whih the plated assigmet φ ad the Majority Vote maj agree ad whih remais assiged durig the uassigmet phase. Seodly we show o variables o whih φ ad maj disagree survive the uassigmet phase. 27

28 For the sake of aalysis of the first part, osider exatly the same uassigmet proess as desribed i Alg3 i whih prior to the uassigmet all variables o whih φ ad maj disagree are uassiged as well (deote this set by A). Clearly, suh a proess a result i oly fewer assiged variables tha the origial proess, ad therefore it is eough to idetify a set as desribed above for the ew proess. Let also B be the set of variables whih support less tha d 1 /2 lauses w.r.t maj ad let C be the set of variables removed durig iteratios. Let S be the set of variables whih remai assiged at the ed of the ew defied proess, S = A B C. Lemma 4.1 guaratees that A 2 Ω(d 2). Also, B 2 Ω(d 1) by the followig argumet: it is eough to osider both variables whih support less tha d 1 /2 lauses w.r.t φ ad those whih appear i some lause together with a variable o whih maj ad φ disagree. Ay idividual variable x supports a total umber of 2 lauses ad eah suh lauses is atually iluded i the formula with probability p 1 = d 1 / 2, so x is expeted to support exatly d 1 lauses (w.r.t φ). The umber of lauses x supports is a biomial radom variable ad strogly oetrated aroud its mea; hee x has probability 2 Ω(d 1) of supportig less tha d 1 /2 lauses. The expeted umber of variables whih support less tha d 1 /2 lauses eah is therefore 2 Ω(d 1) ad by osiderig the martigale of the lause seletio proess the atual umber is ideed 2 Ω(d 1) whp (similarly to Lemma 4.1). I additio, as explaied i Lemma 4.2, the umber of lauses whih otai some variable o whih maj ad φ disagree is whp at most 2 Ω(d 2). The total umber of variables that appear i suh umber of lauses is at most 3 2 Ω(d 2) = 2 Ω(d 2). To summarize, whp there are at most 2 Ω(d 1) + 2 Ω(d 2) 2 Ω(d 1) variables whih support less tha d 1 /2 lauses eah. Assume towards a otraditio that at some iteratio C has reahed the size of A + B ad osider the formula graph idued by the variables of A, B ad C o that iteratio. By our assumptio it has 2( A + B ) 2(2 Ω(d 2) + 2 Ω(d 1) ) 2 Ω(d 1) < O(d 2 1 ) verties. O the other had, the average degree depeds o d 1 sie eah variable supports (w.r.t maj) at least (1/2 1/3)d 1 = d 1 /6 lauses with variables from A B C. All these edges orrespod to 1-lauses also w.r.t φ, exept for maybe 2 Ω(d 2) 2- ad 3-lauses w.r.t φ whih are 1-lauses w.r.t maj, as metioed above. The total otributio of these lauses is egligible sie their umber is muh smaller tha the umber of the variables ivolved ad 28

29 hee we a igore them. Therefore, the average degree is still Ω(d 1 ) with all edges orrespodig to 1-lauses, otraditig Lemma 4.6. We olude that C A + B 2 Ω(d1) whih implies that S 2 C (1 2 Ω(d1) ). For the seod part, by Lemma 4.1 whp there are at most 2 Ω(d2) variables o whih φ ad maj disagree. Cosider the set O osistig of suh variables that have also survived the uassigmet. The eah variable of O must support at least d 1 /3 fully assiged lauses, otherwise it would have bee uassiged durig the iterative proess of the uassigmet (the support is w.r.t σ, the fial assigmet of the uassigmet phase; sie it is partial to maj ad these lauses are fully assiged by σ the support is w.r.t maj as well). I eah suh lause there is aother variable of O sie it is a 1-lause w.r.t maj ad the satisfyig variable is assiged differetly tha i φ. Cosider the formula graph idued by the variables of O. It is of size at most 2 Ω(d2) with average degree at least d 1 /3, i otraditio to Lemma Completig the partial assigmet Lemma 4.9. Let F be a plated radom 3SAT istae as i Theorem 3. The whp over the hoie of F, at the ed of the Small-Support Uassigmet phase of Alg3 the formula graph idued by the uassiged variables has oeted ompoets of size at most log. Proof. Our objetive is to estimate the probability that the formula graph idued by the set of uassiged variables (durig the uassigmet phase) otais a oeted ompoet of size log. Towards estimatig this, we start from a fixed set T of log variables ad ask what is the probability that the followig two evets ourred simultaeously: 1. T has bee all uassiged by the u assigmet proess. 2. The formula graph idued by T is oeted, or, if we rephrase it, the formula otais a set of lauses suh that the orrespodig formula graph idued by T is oeted. I fat, it is eough to osider a miimal set of suh lauses I (i this otext we say a set of lauses is miimal if deletig ay lause disoets the orrespodig subgraph). Two types of lauses are to be osidered: type 1 lauses, whih otai two variables from T, ad type 2 lauses, whih otai three variables from T. We thik of seletig a set of lauses that would idue 29

30 a oeted subgraph o T as the followig proess: we begi with a subgraph otaiig the variables of T but o edges ad add oe lause at a time. The ay type 1 lause a redue the umber of oeted ompoets by at most 1, whereas type 2 redues it by at most 2 (for a miimal set we obtai exatly 1 ad 2, respetively 1 ). Let t deote the umber of variables i T ad t i deote the umber of type i lauses. I the iitial state of the subgraph the umber of oeted ompoets is t ad i the fial state this umber is 1, whih gives us the followig ostrait: t 1 + 2t 2 = t 1. First we aalyse the probability of the seod evet to our for ay set T of size t; deote this probability by P t. P t ( ) ( )( ) ( 7t P r[i 2 7t 3 d2 F ] t t T,I t 1 +2t 2 =t 1 1 t 2 2 ) t1 +t 2 where ( t) is the umber of possibilities to hoose t variables out of the total ; ( ) 7t 2 t 1 is the umber of possibilities to hoose type 1 lauses (satisfied by the plated assigmet φ); ( ) 7t 3 t 2 is the umber of possibilities to hoose type 2 lauses (satisfied by φ); ad ( ) d t1 2 +t 2 is the probability that all hose lauses are atually 2 iluded i the formula. The ases t 1 = 0 ad t 2 = 0 are simpler, thus we perform 1 I fat eve i a miimal set there might be type 2 lauses reduig the umber of oeted ompoets oly by 1. We may thik of suh lauses as type-1 lauses havig their third variable i T ad therefore suh ases are also treated by our alulatio. 30

31 the aalysis assumig either t 1 or t 2 are 0. ( e P t t ( e = t ( e t ( ) 7et 2 t1 ( ) 7et 3 t2 ( ) t1 +t 2 d2 = ) t t 3 t 1 =1 ) t 3 t t 1 t 2 2 ( ) t1 +2t (7ed 2 ) t 1+t 2 t 2 ( ) t1 ( ) t2 t t t 1 =1 ) t 3 t (7e 2 d 2 ) t ( t t 1 ( ) t 1 ( ) t1 ( ) t2 t t t (7ed 2 ) t t t 1 =1 1 t 2 )( ) t (7e 2 d 2 ) t 2 t 2 t = = (28e 2 d 2 ) t. t 2 Bak to the first evet: let S deote the set of variables whih have survived the uassigmet, to be osistet with the otatio of Lemma 4.8 (ad so S deotes the set of variables whih have bee uassiged). From this lemma the probability that all variables of T have bee uassiged is 2 Ω(d 1)t. Let I be a set of lauses as before. I geeral it holds that P r[(i F ) (T S)] = P r[(i F )]P r[(t S) (I F )] T,I T,I = P t P r[(t S) (I F )] If the two evets were idepedet, it would hold that P r[(t S) (I F )] P r[(t S)], hee the desired probability (of both evets to our) would be at most P t P r[t S] (28e 2 d 2 ) t 2 Ω(d 1)t, so for t = log, large eough d 1 ad d 2 2 O(d 1), it ould be upper-bouded by 2 Ω(log ) = o(1) whih ompletes the proof. However, this is ot the ase ad our strategy would be to osider a partiular T T ad a slightly modified uassigmet proess resultig i a set S for whih the two evets would ideed be idepedet. Also ote that for ay T T it holds that P r[(t S) (I F )] P r[(t S) (I F )], hee it is suffiiet to show that for a partiular suh T (defied below) it is true that P r[(t S) (I F )] P r[(t S )]. t 1 t 2 31

32 Let agai I be a miimal set of lauses, the the umber of variables of T that appear at most 6 times i I is at least t/2 (otherwise there would be at least 6t = t lauses i 3 2 I i otraditio to miimality). Deote this subset of variables of T by T. For the sake of aalysis, osider a uassigmet proess as suggested i Lemma 4.8 where A otais ot oly the variables o whih the Majority Vote maj ad the plated assigmet φ disagree, but also those variables o whih the two assigmets agree but the bias of maj towards their assigmet i φ is at most 6. I additio, the followig two sets would be removed prior to the iteratios as well: 1. T \T (whih otais at most t/2 variables); 2. every variable ot i T whih appears more tha 6 times i I (there are at most t/6 suh variables sie I has at most t lauses ad every lause has at most oe variable ot i T ). Deote by S the set of variables whih have survived the modified proess. Whe we would like to emphasize that the survival set S (or S) is defied w.r.t a partiular set of lauses I, we deote it by S I (or S I). I the followig aalysis I deotes the set of lauses of the formula F. Claim 1. S I S I I Proof. First, we show that this relatio holds prior to the iteratios. Let x be a variable i S I I (prior to the iteratios), we will show that x is also i S I. We distiguish two ases: if x T \ T or ot i T but appears more tha 6 times i I the it is i S I by the defiitio of the ew proess; if x / T \ T or ot i T ad appears at most 6 times i I the agai we have two ases: if x is uassiged beause it supports less tha d 1 /2 lauses w.r.t I I the all the more it would support less tha d 1 /2 lauses w.r.t I. If it is uassiged beause maj ad φ disagree o its assigmet, removig the lauses of I i whih it appears at most 6 times a result i a bias of at most 6 towards its assigmet i φ ad hee it is also i S F. We otie that this already implies the assertio of the laim sie the iteratios are defied equally for both proesses. Next, let F = I deote the evet: I is the set of lauses of the formula F. 32

33 We have: P r[(t S )] = P r[f = I] I:T S I P r[f = I] I:T S I I P r[f = I I ] = I :I I =,T S I I = P r[(t S) (I F )] where the first iequality is due to the laim. It remais to boud the probability P r[(t S )]. Ideed, w.r.t every hoie of S (whih depeds o T \T ad T, i.e., it depeds oly o T ) a very similar argumet to that of Lemma 4.8 guaratees that S is of size at least 2 Ω(d1), whp (we otie that prior to the iteratios oly additioal O(log ) variables ould have bee uassiged). Sie this hoie does ot deped o T, we olude that P r[(t S )] 2 Ω(d 1 log ), whih ompletes the proof. 4.5 Proofs of the theorems Proof of Theorem 1. Lemma 4.2 guaratees that durig its approximatio phase, Alg1 will fid a assigmet w.r.t whih the umber of 0- ad 1-lauses i F is at most (ad so the rest are 2- are 3-lauses). Lemma 4.7 implies O(d log d) that i suh ase the umber of uassiged lauses at the ed of the 0-Clause Uassigmet phase of Alg1 is whp at most. Next, Lemma 4.4 guaratees O(d) that whp every subset of this set of lauses is 1-expadig. Now by Hall s oditio there is a mathig of lauses to variables, whih meas that Alg1 would fid a mathig i the bipartite graph ostruted durig its Mathig phase (without spoilig the rest of the lauses whih are already satisfied by other variables). Proof of Theorem 2. Similar to the proof of Theorem 1, substitutig Lemma 4.2 by Lemma 4.3. Proof of Theorem 3. The Majority Vote omputed by Alg3 durig its approxima- 33

34 tio phase agrees with the plated assigmet o at least (1 2 Ω(d2) ) variables whp, as promised by Lemma 4.1. Whe the Small-Support Uassigmet phase begis with suh a assigmet, Lemma 4.8 implies that whp the partial assigmet obtaied at the ed of that phase agrees with the plated assigmet ad that at least (1 2 Ω(d1) ) variables remai assiged. I suh ase we are guarateed the residual formula (whih is idued by the uassiged variables) is ideed satisfiable. The by Lemma 4.9 the oeted ompoets of this formula s graph are whp of size at most log, whih guaratees that Alg3 would omplete the assigmet suessfully i polyomial time durig its exhaustive searh phase. Referees [1] A. Agarwal, M. Charikar, K. Makaryhev, Y. Makaryhev. O( log) approximatio algorithms for Mi UCut, Mi 2CNF Deletio, ad direted ut problems. STOC, [2] N. Alo ad U. Feige. O the power of two, three ad four probes. SODA, , [3] N. Alo ad N. Kahale. A spetral tehiques for olorig radom 3-olorable graphs. SIAM Joural of omputatio 26(6), , [4] N. Alo, M. Krivelevih ad B. Sudakov, Fidig a large hidde lique i a radom graph, Radom Strutures ad Algorithms 13, , [5] E. Be-Sasso, Y. Bilu ad D. Gutfreud. Fidig a Radomly Plated Assigmet i a Radom 3-CNF. Mausript, [6] A. Blum, ad J. Speer. Colorig radom ad semiradom k-olorable graphs. Joural of Algorithms 19, , [7] R. B. Boppaa, Eigevalues ad graph bisetio: a average ase aalysis. FOCS, ,

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