On the sampling Lovász Local Lemma for atomic constraint satisfaction problems
aa r X i v : . [ c s . D S ] F e b ON THE SAMPLING LOVÁSZ LOCAL LEMMA FOR ATOMIC CONSTRAINTSATISFACTION PROBLEMS
VISHESH JAIN, HUY TUAN PHAM, AND THUY DUONG VUONG
Abstract.
We study the problem of sampling an approximately uniformly random satisfyingassignment for atomic constraint satisfaction problems i.e. where each constraint is violated by onlyone assignment to its variables. Let p denote the maximum probability of violation of any constraintand let ∆ denote the maximum degree of the line graph of the constraints.Our main result is a nearly-linear (in the number of variables) time algorithm for this problem,which is valid in a Lovász local lemma type regime that is considerably less restrictive compared toprevious works. In particular, we provide sampling algorithms for the uniform distribution on: • q -colorings of k -uniform hypergraphs with ∆ . q ( k − / o q (1) . The exponent / improves the previously best-known / in the case q, ∆ = O (1) [Jain,Pham, Vuong; arXiv, 2020] and / in the general case [Feng, He, Yin; STOC 2021]. • Satisfying assignments of Boolean k -CNF formulas with ∆ . k/ . . The constant . in the exponent improves the previously best-known in the case k = O (1) [Jain, Pham, Vuong; arXiv, 2020] and in the general case [Feng, He, Yin; STOC 2021]. • Satisfying assignments of general atomic constraint satisfaction problems with p · ∆ . . . The constant . improves upon the previously best-known constant of [Feng, He, Yin;STOC 2021].At the heart of our analysis is a novel information-percolation type argument for showing the rapidmixing of the Glauber dynamics for a carefully constructed projection of the uniform distributionon satisfying assignments. Notably, there is no natural partial order on the space, and we believethat the techniques developed for the analysis may be of independent interest. Introduction
Let X , . . . , X n denote a collection of independent random variables and let C = { C , . . . , C m } denote a collection of events depending on X , . . . , X n (here, the letter C is chosen to represent a“constraint”). For C ∈ C , let vbl( C ) ⊆ { X , . . . , X n } be such that C depends only on X i ∈ vbl( C ) .The celebrated Lovász Local Lemma (LLL) [EL73] (stated here in its variable version, symmetricform) asserts that e · p · ∆ ≤ ⇒ P [ ∧ i ∈ [ m ] C i ] ≥ (1 − e · p ) |C| > , (1.1)where C denotes the complement of the event C , e is the base of the natural logarithm, p = max i ∈ [ m ] P [ C i ] , (1.2)and ∆ ≥ satisfies { j ∈ [ m ] : vbl( C j ) ∩ vbl( C i ) = ∅} ≤ ∆ for all i ∈ [ m ] . (1.3)The original proof of (1.1) is non-constructive and does not provide an efficient algorithm to finda point in ∧ i ∈ [ m ] C i (such a point is called a satisfying assignment). After much work over a periodof two decades (cf. [Bec91, Alo91, MR98, CS00, Sri08, Mos08, Mos09]), the landmark work of Moser nd Tardos [MT10] provided an efficient (randomized) algorithm to find a satisfying assignmentwhenever the LLL condition (i.e. the condition on the left hand side of (1.1)) is satisfied, providedthat one is able to efficiently sample from the distribution of X i , and efficiently able to determinethe set of constraints that are violated by a given realization of X , . . . , X n .In recent years, much attention (cf. [HSZ19, Moi19, GLLZ19, GJL19, FGYZ20, FHY21, JPV20])has been devoted to approximate counting and sampling variants of the algorithmic LLL: underconditions similar to the LLL condition, can we efficiently approximately count the total number ofsatisfying assignments? Can we efficiently sample from approximately the uniform distribution onsatisfying assignments?This problem turns out to be computationally harder than the problem of efficiently finding onesatisfying assignment. Indeed, consider the Boolean k - CNF-SAT problem, in which we are given n Boolean variables x , . . . , x n and m constraints C , . . . , C m such that each constraint depends onexactly k variables, and each constraint is violated by exactly one assignment (out of the k possibleassignments) to its variables. A direct application of the LLL shows that if each constraint sharesvariables with at most (approximately) k /e other constraints, then the formula has a satisfyingassignment, in which case, the algorithm of [MT10] efficiently finds such a satisfying assignment.However, it was shown by Bezáková et al. [BGG +
19] that it is NP -hard to approximately count thenumber of satisfying assignments for a Boolean k -CNF formula in which every variable is allowedto be present in ≥ · k/ constraints, even when the formula is monotone.On the algorithmic side, both deterministic and randomized algorithms have been devised forapproximate counting under LLL-like conditions. On the deterministic side, Moitra [Moi19] pro-vided a deterministic algorithm to approximately count the number of satisfying assignments of aBoolean k -CNF formula in which each constraint shares variables with at most ∆ . k/ otherconstraints (the . hides polynomial factors in k ), provided that k = O (1) . Moitra’s method wasextended by Guo, Liao, Lu, and Zhang [GLLZ19] to provide an efficient deterministic algorithmfor approximately counting proper q -colorings of k -uniform hypergraphs with maximum degree d ,provided that q & d / ( k − and q, k = O (1) . Recently [JPV20], the authors of this paper showedthat for any instance of the variable version, symmetric form of the LLL, if each constraint dependson at most k variables and if each variable takes on at most q values, then there is an efficientdeterministic algorithm to approximately count the number of satisfying assignments provided that q, k = O (1) and p ∆ . , where . hides polynomial factors in q, k . Here, p and ∆ are as in (1.2)and (1.3). In particular, this subsumes and improves upon [Moi19, GLLZ19]. We note that theseapproximate counting algorithms also lead to efficient algorithms for sampling from approximatelythe uniform distribution on satisfying assignments.On the randomized side, algorithms have been devised for instances of the variable version, sym-metric LLL with atomic constraints. Here, an atomic constraint refers to a constraint which isviolated by exactly one assignment to its variables. For the special case of monotone Boolean k -CNF formulas, Hermon, Sly, and Zhang [HSZ19] showed that the Glauber dynamics mix rapidlyprovided that each variable is present in at most c k/ constraints, for some absolute constant c ;note that this matches the hardness regime from [BGG +
19] up to a constant factor. For extremal
Boolean k -CNF formulas (see [GJL19] for the definition) and for d in the entire LLL regime, themethod of partial rejection sampling due to Guo, Jerrum, and Liu [GJL19] allows efficient perfect sampling from the uniform distribution on satisfying assignments. For general Boolean k -CNF for-mulas, Feng, Guo, Yin, and Zhang [FGYZ20] analyzed the Glauber dynamics on a certain “projectedspace” inspired by Moitra’s method, and obtained a near-linear time algorithm for sampling fromapproximately the uniform distribution on satisfying assignments provided that ∆ . k/ ( . hides olynomial factors in k ) – the motivation for constructing the projected space is that, while theoriginal space of satisfying assignments might not even be connected, by passing to an appropriateprojection, not only do we have connectivity, but no bottlenecks for the Glauber dynamics. At thesame time, since we want to be able to sample from the original distribution conditioned on therealisation of the projected assignment, the projection should be relatively ‘mild’ so as not to losetoo much information.Most relevant to this paper is the recent work of Feng, He, and Yin [FHY21], which introducedthe idea of ‘states compression’, thereby considerably expanding the applicability of the methodused in [FGYZ20]. We will survey their results in the next subsection when we compare themwith our own. Here, we only note that compared to the deterministic algorithms for approximatecounting, the randomized algorithms discussed here have two advantages: the running time is muchfaster (in fact, nearly linear in n ), and they are efficient even when parameters such as k, q growwith n . On the other hand, the disadvantage is that so far, these methods are limited to atomicconstraints, whereas the algorithm of [JPV20] is applicable to general instances of the symmetricLLL.1.1. Our results.
We provide randomized algorithms for approximately counting the number ofsatisfying assignments and sampling from approximately the uniform distribution on satisfyingassignments for LLL instances with atomic constraints.We begin with our result for the following class of instances, which capture many interestingproblems such as Boolean k - CNF-SAT and k -hypergraph q -coloring. Later, in Theorem 1.5, wediscuss a result for general atomic constraints. Definition 1.1. A ( k, ∆ , q ) -CSP (constraint satisfaction problem) is an instance of the variableversion, symmetric LLL in which each variable X i is uniformly distributed on an alphabet Ω i ofsize q , each constraint depends on exactly k variables, and each constraint shares variables with atmost ∆ other constraints.As before, we say that a ( k, ∆ , q ) -CSP is atomic if every constraint is violated by exactly oneassignment to its variables. Note that for an atomic ( k, ∆ , q ) -CSP, the LLL asserts that if ∆ ≤ cq k ,for an absolute constant c , then there exists a satisfying assignment. Theorem 1.2.
Given an atomic ( k, ∆ , q ) -CSP on the variables X , . . . , X n , an error parameter ǫ ∈ (0 , / , and a parameter η ∈ (0 , , suppose that one of the following conditions holds.(T1) k ≥ , q ≥ q ( η ) , and ∆ ≤ c ( η ) · q ( k − / o q (1) .(T2) k = 2 , q ≥ q ( η ) , and ∆ ≤ c ( η ) · q / o q (1) .(T3) k ≥ , q ≥ , ∆ ≤ c ( η ) · q . k / ( k · q log q ) .(T4) k ≥ , q = 3 , ∆ ≤ c ( η ) · . k /k .(T5) k ≥ , q = 2 . ∆ ≤ c ( η ) · . k /k .Here q ( η ) and c ( η ) are constants depending only on η . Then, there is a randomized algorithm whichruns in time ˜ O ( n · (( n/ǫ ) η + ∆) · ∆ · k ) , where ˜ O hides polylogarithmic factors in n, ∆ , /ǫ , k , q , and outputs a random assignment X ∈ Q i ∈ [ n ] Ω i such that the distribution µ alg of X satisfies d TV ( µ alg , uniform-satisfying) ≤ ǫ, where uniform-satisfying denotes the uniform distribution on satisfying assignments and d TV denotesthe total variation distance between probability measures. emark. In [FHY21], analogs of (T1) and (T5) are considered. In these cases, they obtain analgorithm for sampling from approximately the distribution uniform-satisfying , and with a similarrunning time, under the more restrictive conditions:(T’1) [FHY21, Theorem 5,4] k ≥ , q ≥ q , and ∆ ≤ q ( k − / .(T’5) [FHY21, Theorem 5.5] k ≥ , q = 2 , ∆ ≤ c ( η ) · k/ . Remark.
We find case (T1) of Theorem 1.2 remarkable since, prior to the work of Moser [Mos08],the best-known version of the existential algorithmic LLL due to Srinivasan [Sri08] required thecondition p ∆ ≤ c (for an absolute constant c , and with notation as in (1.2), (1.3)); in particular,[Sri08] does not guarantee efficiently finding even a single satsifying assignment in the regime (T1)(for sufficiently large q ). The chief innovation of Moser was to use denser witness trees instead ofso-called -trees (Definition 4.3); however, in our work, we are able to bypass the ∆ barrier foratomic ( k, ∆ , q ) -CSPs, for sufficiently large q , even while using -trees.We pause here to record a couple of particularly interesting corollaries of Theorem 1.2. Let H = ( V, E ) denote a k -uniform hypergraph with vertex set V and edge set E . Recall that a proper q -coloring of H is an assignment χ : V → [ q ] such that for every edge e , there exist u, v ∈ e with χ ( u ) = χ ( v ) . In words, no edge is monochromatic. The problem of properly q -coloring H can berecast as an atomic ( k, ∆ · q, q ) -CSP, where ∆ denotes the maximum number of edges that any edgeof H intersects. Indeed, we simply add q constraints for each edge, where constraint i for the edgeis violated if each vertex in the edge is colored with i . Then, by (T1), we have: Corollary 1.3.
Let H = ( V, E ) be a k -uniform hypergraph with k ≥ and let ∆ be defined asabove. Then, for any ǫ, η ∈ (0 , , for q ≥ q ( η ) , and for ∆ ≤ c ( η ) · q ( k − / o q (1) , we can samplefrom a distribution which is ǫ -close in total variation distance to the uniform distribution on proper q -colorings of H , in time ˜ O ( n · (( n/ǫ ) η + ∆) · ∆ · k ) .Remark. This corollary improves upon [FHY21, Theorem 1.3] which requires ∆ ≤ q ( k − / o q (1) ,and on the previous best known regime (even in the bounded degree case) of ∆ . q ( k − / due to[JPV20].The next corollary follows from (T5). Corollary 1.4.
Consider a Boolean k - CNF-SAT instance on n variables x , . . . , x n such that eachconstraint shares variables with at most ∆ other constraints. Then, for any ǫ, η ∈ (0 , and for ∆ ≤ c ( η ) · . k /k , we can sample from a distribution which is ǫ -close in total variation distanceto the uniform distribution on satisfying assignments, in time ˜ O ( n · (( n/ǫ ) η + ∆) · ∆ · k ) .Remark. The constant . in the exponent is within a factor of less than of the hardness regimefrom [BGG + . from [FHY21, Theorem 1.4] and on theprevious best known constant (even in the bounded degree case) of . due to [JPV20].We now present our result for general atomic instances of the LLL. Theorem 1.5.
Given an atomic instance of the LLL, let k denote an upper bound on the numberof variables in any constraint, and let q denote an upper bound on the size of the support of anyvariable X i . Let ǫ, η ∈ (0 , . Let ∆ be as in (1.3) , p ≤ p ( η ) be as in (1.2) , and suppose that p · ∆ . o p (1) ≤ . Then, there is an algorithm which runs in time ˜ O ( n · (( n/ǫ ) η + ∆) · ∆ · k ) , here ˜ O hides polylogarithmic factors in n, ∆ , /ǫ, k, q , and outputs a random assignment X suchthat the distribution µ alg of X satisfies d TV ( µ alg , uniform-satisfying) ≤ ǫ. Remark.
The constant . (which has not been completely optimized and may be slightly lowered)improves upon the constant from [FHY21, Theorem 1.1]. Moreover, given a CSP for which everyconstraint is violated by at most N assignments to its variables, we can construct an atomic CSPwith at most N atomic constraints for every original constraint, and thereby obtain a result similarto Theorem 1.5, with ∆ replaced by ∆ N . We further note that the constant . is almost thesame as the constant in [JPV20]; while Theorem 1.5 only applies to atomic CSPs, its advantageis the much faster running time, as well as an LLL type condition which does not depend on k or q .1.2. Approximate counting.
Theorems 1.2 and 1.5 also imply efficient algorithms for approxi-mately counting the number of satisfying assignments in the same regime. Indeed, by using thesimulated annealing reduction in [FGYZ20], one can easily show that if T ( ǫ ) is the time to obtainone sample (from a distribution which is ǫ -close in total variation distance to the uniform distri-bution), then for any δ ∈ (0 , , there is a randomized algorithm for approximately counting thenumber of satisfying assignments within a multiplicative factor of (1 + δ ) , which runs in time ˜ O (cid:16) mδ T ( ǫ m,δ ) (cid:17) , where m denotes the number of constraints i.e. m = |C| , and ǫ m,δ = Θ (cid:18) δ m log( m/δ ) (cid:19) . Techniques.
In [HSZ19], the authors showed that for a Boolean k -CNF formula which is monotone , the Glauber dynamics on the space of satisfying assignments mix rapidly outside (aconstant factor of) the hardness regime identified in [BGG + k -CNF Boolean formulas, the space of satisfying assignments may not even be connected. Toovercome this barrier, and inspired by Moitra’s approach [Moi19] of ‘marking’ variables, the work[FGYZ20] introduced the following two step procedure for sampling a uniformly random satisfyingassignment: first, sample from the induced distribution on the so-called unmarked variables, andthen, given such a sample Y , sample from the uniform distribution on the satisfying assignmentsconditioned on the assignment to the unmarked variables being Y . The reason the last step iseasy is that, given a typical assignment to the unmarked variables, the remainder of the formulafactors into logarithmic-sized connected components, so that ordinary rejection sampling succeedswith high probability. The key, therefore, is to sample from the induced distribution on unmarkedvariables.The recent work [FHY21] introduced the idea of ‘states-compression’, which generalizes the mark-ing procedure of Moitra. Now, for each variable v with domain Ω v , one constructs a suitable map π v : Ω v → Q v , and the assignment Y now lives in Q v ∈ V Q v . Once again, the projection is to bechosen so that given a typical realisation of Y , the remainder of the formula factors into logarithmic-sized connected components on which ordinary rejection sampling succeeds with high probability,and the main part is showing that one can efficiently sample Y from the corresponding distribution.In order to sample Y , both [FGYZ20,FHY21] show that the Glauber dynamics for the distributionon the space Q v ∈ V Q v , induced by the uniform distribution on satisfying assignments, mix rapidly.For this, both works employ a one-step path coupling argument based on, and extending, the ar-gument from [Moi19]. However, showing that the one-step path coupling is contracting requiresadditional assumptions on the relationship between p and ∆ , and indeed, this is the main reasonfor the degradation of the dependence between p and ∆ in the final results of [FGYZ20, FHY21] and also in [Moi19, GLLZ19]). Furthermore, establishing contraction of the one-step path couplingrequires considerable case analysis for different ranges of the parameters – for instance, the mixingfor the regimes corresponding to Theorem 1.2 and Theorem 1.5 are analyzed separately in [FHY21].The main contribution of our work is to completely dispense with the path coupling analysis of this‘projected Glauber dynamics’ and instead, to devise a novel information-percolation based argumentwhich avoids the need to consider worst-case neighborhoods of a vertex. Such an argument is alsothe crux of [HSZ19] (which, in turn, is inspired by the argument in [LS16]). However, compared to[HSZ19], we critically do not have monotonicity at our disposal. This makes certain ‘sandwiching’arguments inaccessible, and consequently, necessitate developing a careful and somewhat elaboratenotion of combinatorial structures, which we call minimal discrepancy checks (Definition 5.6). Ina nutshell, the information-percolation argument is based on the fact that if the maximal one-step coupling of the Glauber dynamics fails to couple at some time, then there must already beanother discrepancy between the configurations at that time. By tracking the evolution of thesediscrepancies back in time, we show, in fact, that the origin of the failure of the one-step maximalcoupling can be attributed to the appearance of a minimal discrepancy check. By analyzing theseminimal discrepancy checks with considerable care, we then show that they occur with probabilitywhich is essentially just low enough (Proposition 5.9) so as to overcome the union bound on thenumber of possible minimal discrepancy checks. We expect this part of our argument to also beuseful in other contexts.Introducing minimal discrepancy checks allows us to handle the projected Glauber dynamics in allregimes in a unified manner. Another component which facilitates this, and also contributes to ourimproved quantitative estimates, is the notion of admissible projection schemes (Definition 3.2) – incontrast to [FHY21], the conditions we demand of the projections π v : Ω v → Q v are seemingly morecomplicated, but these are exactly the conditions which show up in the analysis of the algorithm,and therefore, avoid unnecessary degradation of the parameters. Once the correct conditions forthe admissible projection schemes are identified (and this is the non-trivial part), the argument forthe existence itself is a standard (although a necessarily quite careful) probabilistic argument.1.4. Organization.
The remainder of this paper is organized as follows. In Section 2, we recordsome preliminary notions related to the Lovász local lemma and constraint satisfaction problems.In Section 3, we introduce admissible projection schemes and state the result (Proposition 3.3)guaranteeing the existence of admissible projection schemes in the LLL regime. The proof ofProposition 3.3, which is initiated in Section 3 is completed in Section 6. In Section 4, we present ourmain sampling algorithm. The main result in this section is Theorem 4.1, which implies Theorem 1.2and Theorem 1.5. The key ingredient required for the proof of Theorem 4.1 is Proposition 4.2. Thisis proved in Section 5, which is the key section of the paper.2.
Preliminaries
Lovász Local Lemma.
The LLL provides a sufficient condition guaranteeing that the prob-ability of avoiding a collection C of “bad events” in a probability space is positive. When the LLLcondition (1.1) is satisfied, the so-called LLL distribution, µ S [ · ] := P [ · | ∧ C ∈C C ] is well-defined (here, the subscript S is chosen to represent “satisfying”). For later use, we recorda standard comparison between the LLL distribution µ S [ · ] and the original distribution P [ · ] on theprobability space i.e. the product distribution on X , . . . , X n . For any event B in the probabilityspace, let Γ( B ) = { C ∈ C : vbl( B ) ∩ vbl( C ) = ∅} . heorem 2.1 (cf. [HSS11, Theorem 2.1]) . Under (1.1) , for any event B in the probability space, µ S [ B ] ≤ P [ B ] Y C ∈ Γ( B ) (1 − e · P [ C ]) − . We also record here the following algorithmic version of the Lovász Local Lemma, which followsdirectly from Moser-Tardos algorithm [MT10] and is also used in [Moi19, FGYZ20, FHY21].
Theorem 2.2 ([MT10]) . Under (1.1) , for any δ ∈ (0 , , there exists a randomized algorithm whichoutputs, with probability at least − δ , a satisfying assignment in time O ( n ∆ k log(1 /δ )) , where k = max C ∈C | vbl( C ) | .Proof. By [MT10], under (1.1), there exists a randomized algorithm which finds a satisfying assign-ment in at most |C| ∆ ≤ n steps in expectation, where each step has time complexity O (∆ k ) . ByMarkov’s inequality, if we run this algorithm for n steps, then with probability at least / , thealgorithm returns a satisfying assignment. The desired conclusion now follows by running log(1 /δ ) independent copies of this algorithm for n steps each. (cid:3) Constraint satisfaction problems.
Let V denote a collection of variables with finite domains (Ω v ) v ∈ V satisfying | Ω v | ≥ for all v ∈ V . A constraint on V is a map C : Y v ∈ V Ω v → { True , False } . We say that C depends on a variable v ∈ V if there exist σ , σ ∈ Q v ∈ V Ω v differing only in v suchthat C ( σ ) = C ( σ ) . For every constraint C , we fix vbl( C ) ⊆ V containing all variables that C depends on. A constraint satisfaction problem (CSP) is specified by Φ = ( V, (Ω v ) v ∈ V , C ) , where C isa collection of constraints. Given a constraint satisfaction problem, we say that σ ∈ Q v Ω v satisfies Φ if and only if C ( σ ) = True for all C ∈ C . We define the degree ∆ of a CSP to be ∆ = max C ∈C |{ C ′ ∈ C : vbl( C ) ∩ vbl( C ′ ) = ∅}| . We say that C is an atomic constraint if | C − (False) | = 1 . A CSP Φ is said to be atomic if every C ∈ C is an atomic constraint. Popular examples of atomicconstraint satisfaction problems are: • k -CNF-SAT. Here, Ω v = { , } for all v ∈ V and | vbl( C ) | = k for all C ∈ C . • k -Hypergraph q -coloring. Let H = ( V, E ) denote a k -uniform hypergraph. To each vertex v ∈ V , we assign a color in [ q ] such that no hyperedge is monochromatic. This correspondsnaturally to an atomic CSP Φ = ( V, (Ω v ) v ∈ V , C ) with Ω v = [ q ] for all v ∈ V and C = { C e,i : e ∈ E , i ∈ [ q ] } where for σ ∈ [ q ] V , C e,i ( σ ) = False ⇐⇒ σ ( w ) = i ∀ w ∈ e. To every CSP, we associate an instance of the LLL as follows: the random variables are X , . . . , X v ,where each X i is uniformly distributed on Ω i . To each constraint C ∈ C , we associate the event ( σ ∈ Y v ∈ V Ω v : C ( σ ) = False ) . We will abuse notation and denote this event by C and the collection of all such events by C . inally, for a CSP Φ , we let µ Φ denote the LLL distribution of the associated LLL instance i.e. µ Φ is the uniform distribution on satisfying assignments of Φ . When the underlying CSP is clearfrom context, we will omit the subscript and denote µ Φ simply by µ .3. Projection schemes
Preliminaries.
Given a CSP
Φ = ( V, (Ω v ) v ∈ V , C ) , a projection scheme is a collections of maps π v : Ω v → Q v , where Q v is a finite alphabet with | Q v | ≥ . We will frequently denote the collection ( π v ) v ∈ V simply by π . We let P π denote the product distribution on Q v ∈ V Q v induced via π by the uniformdistribution on Q v ∈ V Ω v . We also let µ π denote the distribution on Q v ∈ V Q v induced via π by µ = µ Φ .Let Φ be an atomic CSP. Recall that this means that for each C ∈ C , there exists some C ∈ Q v ∈ vbl( C ) Ω v such that X ∈ Q v ∈ V Ω v does not satisfy C if and only if X ( v ) = C ( v ) ∀ v ∈ vbl( C ) . Given an atomic CSP Φ and a projection scheme π , for every C ∈ C , we define C π ∈ Q v ∈ vbl( C ) Q v by C π ( v ) = π v ( C ( v )) ∀ v ∈ vbl( C ) . This naturally leads to a CSP Φ π = ( V, ( Q v ) v ∈ V , C π ) , where for each C ∈ C , there is a constraint C π ∈ C π such that for Y ∈ Q v ∈ V Q v , C π ( Y ) = False ⇐⇒ Y ( v ) = C π ( v ) ∀ v ∈ vbl( C ) . Motivated by this, for a constraint C π ∈ C π , v ∈ vbl( C π ) := vbl( C ) and Y ∈ Q v ∈ V Q v , we say that Y ( v ) does not satisfy C π if and only if Y ( v ) = C π ( v ) .For a constraint C ∈ C , let b ( C ) := max Y ∈ Q v ∈ V Q v Y u ∈ vbl( C ) P [ X ( u ) = C ( u ) | Y ] , where Y = π ( X ) . Equivalently, b ( C ) = Y u ∈ vbl( C ) | π − u ( C π ( u )) | − . Let b := max C ∈C b ( C ) . Also, let q := max v ∈ V,Y ∈ Q u ∈ V Q u d TV ( P π [value( v ) = · ] , µ π [value( v ) = · | Y − v ]) , where Y − v denotes the | V | − dimensional vector obtained by removing Y ( v ) from Y .The following useful bound on the conditional marginals of µ π follows from Theorem 2.1. Lemma 3.1.
Let Φ be an atomic CSP and let π be a projection scheme. Suppose that e · b · ∆ ≤ .Then for any v ∈ V and any partial assignment Z ∈ Q u ∈ V \{ v } Q u , µ π [value( v ) = · | Z ] ≤ (1 − b ) − ∆ P π [value( v ) = · ] . roof. Consider the product distribution P Z on Q v ∈ V Ω v where each coordinate u ∈ V \ { v } is dis-tributed according to P [ X ( u ) = ·| π u ( X ( u )) = Z ( u )] and the v th coordinate is uniformly distributedon Ω v . The P Z probability that a constraint C ∈ C is not satisfied is at most b ( C ) by definition. Let µ Z,S denote the distribution on satisfying assignments of Φ induced by P Z . Then, since e · b · ∆ ≤ ,we have by Theorem 2.1 that µ π [value( v ) = · | Z ] = P [ π v ( X ( v )) = · | π u ( X ( u )) = Z ( u ) , X satisfies all C ∈ C ]= µ Z,S [ π v ( X ( v )) = · ] ≤ P Z [ π v ( X ( v )) = · ] Y C ∈C : v ∈ vbl( C ) (1 − e · P Z [ C ]) − ≤ P π [value( v ) = · ](1 − b ) − ∆ . (cid:3) Let Φ be an atomic CSP and let π be a projection scheme. For each constraint C ∈ C , let vbl( C ) denote the set of variables v in C for which | Q v | > . Also, for C ∈ C , let ζ ( C ) := max v ∈ vbl( C ) (cid:18) , min (cid:18) (1 − b ) ∆ q P π [value( v ) = C π ( v )] , (cid:19)(cid:19) Admissible projection schemes.
The next definition isolates the class of projection schemeswe will be interested in.
Definition 3.2.
Let Φ be an atomic CSP and let π be a projection scheme. Let η ∈ (0 , / . Wesay that π is admissible if(A1) b ≤ η/ (300∆) .(A2) There exists κ ≥ such that for any C ∈ C , | vbl( C ) | · κ · ζ ( C ) · Y v ∈ vbl( C ) (cid:16) (1 − b ) − ∆ P π [value( v ) = C π ( v )] + e − κ/ (cid:17) ≤ (60000∆) − . Furthermore, κ ≤ K (log ∆ + log q + log k ) for a universal constant K , for q = max v ∈ V | Q v | ,and for k = max C ∈C | vbl( C ) | .(A3) For any v ∈ V and C, C ′ ∈ C with v ∈ vbl( C ) ∩ vbl( C ′ ) , P π [value( v ) = C π ( v )] ≤ P π [value( v ) = C ′ π ( v )] ≤ P π [value( v ) = C π ( v )] . (A4) For v ∈ V , π ( v ) can be computed in time K log | Ω v | , and for any q ∈ Q v a uniform value in π − v ( q ) can be sampled in time K log | Ω v | , where K is a universal constant. Remark.
Note that the condition b ≤ η/ (300∆) for η < / in (A1) guarantees that (1 − b ) − ∆ ≤ b ∆ ≤ η/ . The following is the main result of this section.
Proposition 3.3.
Let
Φ = ( V, (Ω v ) v ∈ V , C ) be an atomic CSP. Suppose that at least one of thefollowing holds:(1) | Ω v | = A ≥ A for all v ∈ V , | vbl( C ) | = k ≥ for all C ∈ C , and ∆ ≤ A g ( k ) − o A (1) , where A is a constant depending only on η , and g ( k ) = max (cid:8) k − , k (cid:9) .(2) | Ω v | = A = 2 for all v ∈ V , | vbl( C ) | = k ≥ for all C ∈ C , and ∆ ≤ cA . k /k where c is a constant depending only on η .(3) | Ω v | = A = 3 for all v ∈ V , | vbl( C ) | = k ≥ for all C ∈ C , and ∆ ≤ cA . k /k where c isa constant depending only on η . | Ω v | = A ≥ for all v ∈ V , | vbl( C ) | = k ≥ for all C ∈ C , and ∆ ≤ cA . k − / ( k log A ) ,where c is a constant depending only on η .(5) ∆ ≤ p − . o p (1) , where p ≤ c for a constant c depending only on η .Then, there exists an admissible projection scheme π = ( π v ) v ∈ V with π v : Ω v → Q v . Moreover, forany δ ∈ (0 , , this projection scheme can be constructed, with probability at least − δ , in time O ( n ∆ k log(1 /δ )) , where k = max C ∈C | vbl( C ) | .Proof. We give here the complete proof of