A constructive proof of the Lovasz Local Lemma
The Lovasz Local Lemma [EL75] is a powerful tool to prove the existence of combinatorial objects meeting a prescribed collection of criteria. The technique can directly be applied to the satisfiability problem, yielding that a k-CNF formula in which …
Authors: Robin A. Moser
A constructiv e pro of of the Lo v´ asz Lo cal Lemma Robin A. Moser ∗ Institute for Theoretical Computer Science Departmen t of Computer Science ETH Z ¨ uric h, 8092 Z ¨ uric h, Switzerland robin.mo ser@inf.ethz. ch Octob er 2008 Abstract The Lov´ asz Local Lemma [EL75] is a powerful to ol to prove the existence of combinatorial ob jects meeting a prescrib ed c o llection of criter ia . The tec hnique can directly be applied to the satisfiabilit y problem, yielding that a k -CNF formula in whic h each clause has common v ariables with at most 2 k − 2 other c la uses is alwa ys satisfiable. A ll hitherto known pro ofs of the Lo cal Lemma are non-cons tructiv e and do th us not provide a r ecipe as to how a satisfying assignment t o such a formula can b e efficiently found. In his br eakthrough pap er [B e c 91], Beck demonstrated that if the neighbour hoo d of ea c h clause be restricted to O (2 k/ 48 ), a polyno mial time algor ithm for the search problem exists. Alon simp lified and randomized his pro cedure and improv ed the bound to O (2 k/ 8 ) [Alo 91]. Sriniv asan presented in [Sri08] a v ariant that achiev es a b ound of essentially O (2 k/ 4 ). In [Mos08], we improv ed this to O (2 k/ 2 ). In the pres en t paper , we give a ra ndomized alg orithm that finds a satisfying a s signmen t to every k -CNF formula in which eac h clause ha s a neig h b ourho od of at most the a symptotic o ptim um of 2 k − 5 − 1 other clauses and that runs in expected time p olynomial in the size of the form ula, irr espective of k . If k is considered a constant, we can a lso give a deterministic v ariant. In contrast to all previous a pproaches, our analysis do es not anymore inv oke the standar d non-constructive versions of the Lo cal Lemma a nd can ther e fo re b e considered an alternative, co nstructiv e pro of of it. Key W ords and Phrases. Lo v´ asz Lo cal Lemma, derandomizat ion, b ound ed o ccurrence SA T in s tance s, h yp ergraph colouring. 1 In tro duction W e use th e notational framew ork in tro duced in [W el08]. W e assume an in fi nite su pply of prop ositional variables . A liter al L is a v ariable x or a complemen ted v ariable ¯ x . A finite set D of literals ov er p airwise distinct v ariables is called a clause . W e sa y that a v ariable x o c curs in D if x ∈ D or ¯ x ∈ D . A fi nite set F of cla uses is call ed a formula in CNF (Conjunctiv e Normal F orm). W e sa y that F is a k -CNF form ula if ev ery clause h as size exactly k . W e write vbl( F ) to denote the s et of all v ariable s o ccurring in F . Lik ewise, vbl( D ) is the set of v ariables o ccurr ing in a clause D . F or a literal L , let vbl( L ) ∈ vbl( F ) refer directly to the underlying v ariable. A truth assignment is a function α : vbl( F ) → { 0 , 1 } wh ic h assigns a b oolean v alue to eac h v ariable. A literal L = x (or L = ¯ x is satisfie d by α if α ( x ) = 1 (or α ( x ) = 0). A clause is satisfie d by α if it con tains ∗ Researc h is supp orted by the SNF G rant 20002 1-118001/1 1 a satisfied lit eral and a formula is satisfie d if all of its clauses are. A formula is satisfiable if there exists a satisfying truth assignmen t to its v ariables. Let k ∈ N and let F b e a k -CNF form ula. The dep endency gr aph of F is d efined as G [ F ] = ( V , E ) with V = F and E = {{ C, D } ⊆ V | C 6 = D , vbl( C ) ∩ vbl( D ) 6 = ∅} . The neighb ourho o d of a gi ve n clause C in F is defin ed as the set Γ F ( C ) := { D | D 6 = C, vbl( D ) ∩ vbl( C ) 6 = ∅} of a ll clauses sharing common v ariables with C . It coincides with the set of verti ces adjac ent to C in G [ F ]. T he inclusive neighb ourho o d of C is defined to b e Γ + F ( C ) := Γ F ( C ) ∪ { C } . Supp ose U ⊆ vbl( F ) is an arb itrary subset of the v ariables o ccurring in F , then we will d enote by F ( U ) := { C ∈ F | vbl( C ) ∩ U 6 = ∅} the sub form ula that is affe c te d by these v ariables. If α is an y assignment for F , we wr ite vlt( F , α ) to denote the set of clauses w hic h are vio lated by α . The Lo v´ asz Lo cal Lemma w as in trod uced in [EL75] as a to ol to pro v e the existence of com binatorial ob jects meeti ng a pr escribed collection of criteria. A simple, symmetric and uniform version of the Lo cal Lemma can b e directly form ulated in terms of satisfiabilit y . It then reads as follo ws. Theorem 1.1. [EL75] If F is a k -CNF formula such that al l of its clauses C ∈ F satisfy the pr op erty | Γ F ( C ) | ≤ 2 k − 2 , then F is satisfiable. The hitherto kno wn pro ofs of this statemen t are non-constructiv e, m eaning that they do not disclose an effici ent ( p olynomial-time) method to find a satisfying a ssignment . Whether t here exists an y su ch metho d was a long-sta nding op en pr oblem unti l Bec k pr esente d in h is breakthrough paper [Be c91] an algorithm that finds a satisfying assignment in p olynomial time, at least if | Γ F ( C ) | ≤ 2 k / 48 . Using v arious guises of the to ols in tro duced by Bec k, there h a v e b een sev eral attempts to impro v e up on the exp onen t [Alo91, Mos06, S ri08, Mos0 8 ], but a significan t gap alw a ys r emai ned. In the pr esen t pap er w e will close that gap to the asymptotic optim um. While the to ols applied in the analysis are still deriv ed from the original ap p roac h b y Bec k and also significan tly from the randomization and simp lificat ion cont ribu ted b y Alon, the algorithm itself n o w lo oks substant ially differen t. An in teresting n ew asp ect of the present analysis is that the standard non-constructiv e pro ofs of the Lo cal Lemma are not inv ok ed an ymore. The algorithm w e presen t and its pro of of c orrectness can b e therefore co nsidered a constructiv e pr oof of the Lo cal L emma, or at least of its incarnation for satisfiabilit y . W e conjecture that the metho ds prop osed can b e seamlessly translated to most applicatio ns co v ered by the framew ork b y Mollo y and Reed in [MR98], ho w ev er, this remains to b e form ally chec ked. O u r main result will b e the follo wing. Theorem 1.2. If F is a k -CNF formula such that al l clauses C ∈ F satisfy the pr op erty | Γ + F ( C ) | ≤ 2 k − 5 , then F i s satisfiable and ther e exists a r andomize d algorithm that finds a satisfying assignment to F i n exp e cte d time p olynomial in | F | (indep endent of k ). If w e drop the requiremen t that the algorithm b e of p olynomial runn ing time for asymp toti cally growing k , then w e can also derand omize the pro cedure. Theorem 1.3. L et k b e a fixe d c onstant. If F is a k -CNF formula such that al l clauses C ∈ F satisfy the pr op erty | Γ + F ( C ) | ≤ 2 k − 5 , then F is satisfiable and ther e exists a deterministic algorith m that finds a satisfying assignment to F in time p olynom ial in | F | . In the sequel w e shall prov e the tw o claims. 2 A randomized algorit hm based on loc al correcti ons The algorithm is as simple and natural as it can get. Basically , we s tart w ith a r andom assignment , then we c hec k whether an y clauses are violated and if so, w e pic k one of them and sample another r andom assignmen t 2 for the v ariables in that clause. W e con tin ue doing this unti l we either fi nd a satisfying assignment or if the correction pro cedure tak es to o muc h time, we give u p and restart with another random assignment. Th is v ery basic metho d tur ns out to b e suffi cientl y strong for the c ase of form ulas with small neigh b ourho o ds, the only thing we need to do so as to mak e a run ning time analysi s p ossible is to select the clauses to b e corrected in a somewhat systematic fashion. As in the theorem, let F b e a k -CNF form ula o v er n v ariables w ith m clauses, suc h that ∀ C ∈ F : | Γ + F ( C ) | ≤ d , with d := 2 k − 5 . W e imp ose an arbitrary , globally fixed ord ering up on the clauses of F , let us call th is the lexic o gr aphic o r dering . W e d efine a recursive pro cedure which tak es the formula F , a starting assignmen t α and a clause C ∈ F which is violated by α as inpu t, and outpu ts another assignment that arises from α b y p erf orming a series of lo cal corrections in the p r o ximit y of C . function lo cally correct( F, α, C ) α ← α with the a ssignmen ts for vbl( C ) replaced by random v alues (u.a.r.); while vlt(Γ + F ( C ) , α ) 6 = ∅ do D ← lexico graphically first clause in vlt(Γ + F ( C )); α ← lo cally correct( F, α, D ); return α ; Algo rithm 2.1: r e cu rsive pr o c e dur e for lo c al c orr e ctions As y ou immediately notice, this recursion h as the p oten tial of run ning foreve r. W e will h o w ev er see that long r u nning times are unlik ely to o ccur. If we are unlu cky enough to encounte r such a case, w e in terrupt the algorithm prematurely . The follo wing algorithm now uses the describ ed recursive sub p ro cedu re in order to find a satisfying assignmen t for the w hole formula. function solve lll( F ) α ← a n assignment pick ed uniformly at random from { 0 , 1 } vbl ( F ) ; while vlt( F, α ) 6 = ∅ do D ← lexico graphically first clause in vlt( F, α ); α ← lo cally correct( F, α, D ); keep tr a c k of the num b er of r ecursive inv o cations done by lo cally correct; if the n umber exceeds log m + 2 , then ab ort the whole lo op and restart, sampling another α . return α ; Algo rithm 2.2: the c omp lete solver If the algorithm terminates, the r esu lt clea rly constitutes a satisfying assignment. W e ho w ev er ha v e to c hec k that the exp ected run ning time is p olynomia l. T h e r emai nd er of the pro of is to establish this prop ert y . In order to b e able to talk ab out the b eha viour of the alg orithm we need to contro l the rand omness injected. Let us formalize the r andom bits used in a wa y that will greatly simp lify the analysis. Let us say that a total fun ctio n A : vbl( F ) × N 0 → { 0 , 1 } is a table of assignments for F . Let us extend the notion to 3 literals in the natural manner, i.e. A ( ¯ x, i ) := 1 − A ( x, i ). F ur thermore call a total fu nction α : vb l( F ) → N 0 an indir e ct assignment . Give n a fixed table o f assignmen ts, an indirect assignmen t a utomatically induces a standard truth assignment ⋆ α which is defined as ⋆ α ( x ) := A ( x, α ( x )) for all x ∈ vbl( F ). Let us now, just f or the analysis, imagine that the algorithm w orks w ith indirect assignmen ts instead of standard ones. That is, ins tea d of sampling a starting assignment un iformly at rand om, solv e lll( ... ) could samp le a table of assignment s A uniformly at rand om b y randomly selecting eac h of its entries. It will th en hand ov er the pair ( A , α ) to lo cally c orrect( ... ), where α is an indir ect assignmen t this time w hic h tak es zero es everywhere. Note that this is equiv alen t sin ce ⋆ α is u niformly distribu ted. Eac h time the v alue of a v ariable is s u pp osed to b e resampled inside lo cally correct( ... ), w e in s tead increase its indirect v alue by one. Note that this equally completely corresp onds to s amp ling a new r andom v alue since the corresp onding ent ry of the table has n ev er b een u sed b efore. In the s equ el, we will adopt this view of the algorithm a nd its acquisition of randomness. W e n eed to b e able to record an accurate journal of what the algorithm do es. A r e cu rsion tr e e τ is an (un ordered) ro oted tree together with a lab elling σ τ : V ( τ ) → F of eac h vertex with a clause of the form ula su c h that if for u, v ∈ V ( τ ), u is the parent no de of v , then σ τ ( v ) ∈ Γ + F ( σ τ ( u )). W e can record the actions of the recursiv e pro cedure in terms of su c h a recursion tree, where the r oot is lab elled with the clause the p rocedu re was originally called for and all d escendan t no des r epresen ting the recursive in v o cations and carrying as lab els the clauses handed o v er in those. Let n o w a table of assignments b e globally fixed and let α b e any indirect assignmen t and C ∈ F some clause violated u nder ⋆ α . Su pp ose that lo cally correct( ... ) h alts on inputs α and C . Then w e sa y that the c omp lete r e cursion tr e e on the giv en input is the co mplete representat ion of the recursiv e pr ocess up to the p oin t where it returns. Ev en if the pro cess do es not return or do es not return in the time we allot, w e ca n capture the tree represen ting th e recursiv e inv o cations made in an y inte rmediate ste p and w e call these i nterme diate r e c u rsion tr e es . Let τ be any recursion tree. The size | τ | of τ is defined to b e th e n umb er of vertice s. In order to simplify notatio n, we will write [ v ] := σ τ ( v ) for any v ∈ V ( τ ) to denote the lab el o f v ertex v . Let us sa y that a v ariable x ∈ vb l( F ) o c curs in τ if there exists a v ertex v ∈ V ( τ ) suc h that x ∈ vbl([ v ]). W e write vbl( τ ) to denote the set of v ariables th at o ccur in τ . W e are now r eady to mak e a first statemen t ab out the correctness of the algorithm. Lemma 2.1. L et F and A b e glob al ly fixe d. L et α b e any indir e ct assignment and C ∈ F a cla use violate d under ⋆ α . Supp ose lo cally correct( ... ) halt s on input α and C and let τ b e the c omp lete r e cursion tr e e for this i nvo c ation. Then the as signment α ′ which the function outputs satisfies the sub f orm ula F ( vbl ( τ )) . Pr o of. Assume that α ′ violate s an y clause D ∈ F ( vbl ( τ )) . Su pp ose furtherm ore that dur ing the pro cess we ha v e recorded the ordering in whic h th e recursiv e inv o cations w ere made. No w let v ∈ V ( τ ) b e the last v ertex (according to that ordering) of which the corresp onding lab el [ v ] shares common v ariables w ith D . Since an y in v o cation on inp ut [ v ] can only return once all clauses in Γ + F ([ v ]) are satisfied, it cannot return b efore D ∈ Γ + F ([ v ]) is. Since after that n o c hanges of the assignments for the v ariables in D h av e o ccured (b y c hoice of v ), D is still satisfied when the fun ctio n return s, a con tradiction. In the pro of we had to assume ha ving remem b ered th e ordering in whic h the pro cess generated the v ertices. Ho wev er, from the statemen t of the lemma we can no w infer that this is not necessary since that ordering can b e reconstructed by just lo oking at the shap e and th e lab els. F or any recursion tree τ , let us defin e the natur al or dering π τ : V ( τ ) → [ | V ( τ ) | ] to be the ordering we obtain b y starting a depth-first searc h at the r oot and at eve ry no de, selecting among th e not-y et tra v ersed c hildren the one w ith the lexicog raphically first lab el (we u se the notation [ n ] := { 1 , 2 , . . . , n } for n ∈ N ). W e claim the fol lo wing. Lemma 2.2. L et τ b e any (interme diate) r e cursion tr e e pr o duc e d b y any (p ossibly non-terminating) c al l to lo cal ly correct( ... ) . The or dering in which the pr o c ess made r e cursive invo c ations c oincides with the natur al or dering of τ and in p articular , ther e is no no de with two identic al ly lab el le d childr en. 4 Pr o of. Note that a recursive pro cedure naturally generates a r ecursion tree in d epth-first s earch ord er. What w e ha v e to chec k is merely that the c hildren of eve ry no de are generated according to the lexicographic ordering of the their lab els. W e pro ceed by indu ctio n. T he recursion tree τ h as a ro ot r lab elle d [ r ] and a series of c hildren. Assume as ind uction hypothesis that th e subtrees ro oted at th e c hildren hav e the required prop ert y . W e ha ve to pr o v e that on the h ighest inv o cat ion lev el wh ere the lo op chec ks for un satisfied clauses in Γ + F ([ r ]), no clause is either pic ke d t wice or is pic ke d despite the fact that it precedes a clause already pic k ed during the same loop in the lexicographic ordering. Both p ossibilities are excluded by Lemma 2.1, whic h readily implies that after a recursive in v o cation returns, the set of violated clauses is a strict subset of the set of clauses violated b efore and in particular the clause we made the call f or is satisfied eve r after. It can easily b e verified that the fact that w e allo w intermedia te tree s d oes not harm . Lemma 2.1 also implies that the maxim um n umber of ti mes the oute r lo op (in solv e lll( ... )) needs to b e rep eated is b ound ed by O ( m ) since the total num b er of violated cl auses cannot b e any larger and eac h call to the correction pro cedure eliminates at least one. Since w e ab ort the recursion whenev er it tak es more than logarithmically many in v o cations, clearly th e w hole lo op terminates after a p olynomial num b er of steps. The only thing left to pro v e is that it is n ot necessary to ab ort and ju mp bac k to the b eginning more often th an a p olynomial num b er of times. In the sequel, w e will show that in the exp ected case, this has to b e done at m ost twice . 3 Consistency and comp osite witnesses Supp ose that w e fi x an assignment table A and an in direct assignmen t α and we call locally correct( ... ) for some violated clause C and wait. Supp ose that the fun ctio n do es not return w ithin the log ( m ) + 2 steps allotted and we ab ort. Now let τ b e the r ecursion tree that h as b een pr od uced up to the time of in terruption. τ has at least log( m ) + 2 vertic es (up to some rounding issues it has exactly that many). Supp ose that someb o dy w ere to b e con vinced that it wa s really necessary for the pr ocess to tak e that long, then we can present them the tree τ as a justification. By insp ecting τ together with α , they can v erify that for the giv en table A , the correction could not ha ve b een completed any faster. S uc h a certificatio n concept will no w allo w u s to estimate the probabilit y of abortion. W e hav e to introdu ce some formal n otions. Let v ∈ V ( τ ) b e any v ertex and x ∈ vbl([ v ]) a v ariable that o ccurs there. W e define the o c curr enc e index of x in v to b e the num b er of times that x has o ccurred b efore in the tree, written idx τ ( x, v ) := |{ v ′ ∈ V ( τ ) | π τ ( v ′ ) < π τ ( v ) , x ∈ vbl([ v ′ ]) }| . If δ is an y indirect assignment , we say that τ is c onsistent with A offset b y δ if the prop ert y ∀ v ∈ V ( τ ) : ∀ L ∈ [ v ] : A ( L, idx τ (vbl( L ) , v ) + δ ( x )) = 0 holds. F ur thermore, let u s defi n e the offset assignment induc e d by τ as δ τ ( x ) := |{ v ∈ V ( τ ) | x ∈ vb l([ v ]) }| for all x ∈ vb l( F ). Recall n o w w hat the r ecur s ion pr ocedur e do es; it s tarts w ith a give n indirect assignmen t and p erforms recursiv e inv ocations in the natural ordering of th e pro du ced r ecursion tree and in eac h in v o cation, the indirect assignmen ts of the v ariables in the corresp onding clause are incremented. Th is immediately yields the follo wing observ ation. Observ ation 3 .1. If τ is any (interme diate) r e cursion tr e e of an inv o c ation of the r e cursive pr o c e dur e for a ta ble A and an indir e c t sta rting assignment α , then τ is c onsistent with A offset by α . Let a collecti on W := { τ 1 , τ 2 , . . . , τ t } of recursion trees b e giv en of whic h the r oots h av e pairwise distinct lab els. Consider an auxiliary graph H W of whic h those trees are the v ertices and t w o d istinct v ertices τ i , τ j 5 are connected by an edge if the corresp ondin g t w o trees sh are a common v ariable, i.e. vbl( τ i ) ∩ vbl( τ j ) 6 = ∅ . If H W is connected, then W is said to b e a c omp osite witness for F . The vertex set of a comp osite witness is d efined to b e V ( W ) := { ( τ , v ) | τ ∈ W , v ∈ V ( τ ) } , i.e. it is the set of all vertic es in any of th e separate trees, eac h ann ota ted by its tree of origin. T o sh orten notatio n, we acc ess the lab el of su c h a vertex by writing [ w ] := [ v ] f or eac h w = ( τ , v ) ∈ V ( W ). The size of a comp osite witness is defined to b e | V ( W ) | . There is a natur al w a y of tra v ersing the v ertices in V ( W ). Consider that ea c h of the recursion trees τ i has a distinctly labelled r oot. No w order the trees ac cording to the lexicographic orderin g of the r o ot lab els. T rav erse the fir st tree according to its n atural ord ering, then the second one, and so forth. W e call this the natur al or dering for a comp osite witness and we wr ite π W : V ( W ) → [ | V ( W ) | ] to d en ote it. Let v ∈ V ( W ) b e any vertex of the comp osite witness and x ∈ vbl([ v ]) any v ariable that o ccurs there. W e define th e o c curr enc e index of x in v to b e the num b er of times that x has o ccurred b efore in the witness, written idx W ( x, v ) := |{ v ′ ∈ V ( τ ) | π W ( v ′ ) < π W ( v ) , x ∈ vbl([ v ′ ]) }| . W e say th at a comp osite witness is c onsistent with A if the pr op erty ∀ v ∈ V ( W ) : ∀ L ∈ [ v ] : A ( L, idx W (vbl( L ) , v )) = 0 holds. The defin itions immediately imply the follo wing. Observ ation 3.2. If W = { τ 1 , τ 2 , . . . , τ t } is a c omp osite witness with the r e cursion tr e es or der e d ac c or ding to the lexic o gr aphic al or dering of their r o ot lab e ls, then W is c onsist ent with A if and only if for al l 1 ≤ i ≤ t , τ j is c onsistent with A offset by P j
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