Hard constraint satisfaction problems have hard gaps at location 1

Hard onstrain t satisfation problems ha v e hard gaps at lo ation 1 ∗ P eter Jonsson † Andrei Krokhin ‡ F redrik Kuivinen § Mar h 4, 2022 Abstrat An instane of the maximum  onstr aint satisfation pr oblem ( Max CSP ) is a nite olletion of onstrain ts on a set of v ariables, and the goal is to assign v alues to the v ariables that maximises the n um b er of satised onstrain ts. Max CSP aptures man y w ell-kno wn problems (su h as Max k -SA T and Max Cut ) and is onsequen tly NP - hard. Th us, it is natural to study ho w restritions on the allo w ed onstrain t t yp es (or  onstr aint language ) aet the omplexit y and appro ximabilit y of Max CSP . The PCP theorem is equiv alen t to the existene of a onstrain t language for whi h Max CSP has a hard gap at lo ation 1, i.e. it is NP -hard to distinguish b et w een satisable instanes and instanes where at most some onstan t fration of the onstrain ts are satisable. All onstrain t languages, for whi h the CSP problem (i.e., the problem of deiding whether all onstrain ts an b e satised) is urren tly kno wn to b e NP -hard, ha v e a ertain algebrai prop ert y . W e pro v e that an y onstrain t language with this algebrai prop ert y mak es Max CSP ha v e a hard gap at lo ation 1 whi h, in partiular, implies that su h problems annot ha v e a PT AS unless P = NP . W e then apply this result to Max CSP restrited to a single onstrain t t yp e; this lass of problems on tains, for instane, Max Cut and Max DiCut . Assuming P 6 = NP , w e sho w that su h problems do not admit PT AS exept in some trivial ases. Our results hold ev en if the n um b er of o urrenes of ea h v ariable is b ounded b y a onstan t. W e use these results to partially answ er op en questions and strengthen results b y Engebretsen et al. [Theor. Comput. Si., 312 (2004), pp. 17 45℄, F eder et al. [Disrete Math., 307 (2007), pp. 386392℄, Krokhin and Larose [Pro . Priniples and Pratie of Constrain t Programming (2005), pp. 388402℄, and Jonsson and Krokhin [J. Comput. System Si., 73 (2007), pp. 691702℄. Keyw ords: onstrain t satisfation, optimisation, appro ximabilit y , univ ersal algebra, ompu- tational omplexit y , di hotom y ∗ Preliminary v ersions of parts of this rep ort app eared in Pr o  e e dings of the 2nd International Computer Sien e Symp osium in R ussia (CSR-2007) , Ek aterin burg, Russia, 2007 † Departmen t of Computer and Information Siene, Link öpings Univ ersitet, SE-581 83 Link öping, Sw eden, email: petejida.liu.se , phone: +46 13 282415, fax: +46 13 284499 ‡ Departmen t of Computer Siene, Univ ersit y of Durham, Siene Lab oratories, South Road, Durham DH1 3LE, UK, email: andrei.krokhindurham.a. uk , phone: +44 191 334 1743 § Departmen t of Computer and Information Siene, Link öpings Univ ersitet, SE-581 83 Link öping, Sw eden, email: frekuida.liu.se , phone: +46 13 286607, fax: +46 13 284499 1 1 In tro dution Man y om binatorial optimisation problems are NP -hard so there has b een a great in terest in onstruting appro ximation algorithms for su h problems. F or some optimisation prob- lems, there exist p o w erful appro ximation algorithms kno wn as p olynomial-time appr oximation shemes (PT AS). An optimisation problem Π has a PT AS A if, for an y xed rational c > 1 and for an y instane I of Π , A ( I , c ) returns a c -appro ximate (i.e., within c of optim um) solution in time p olynomial in |I | . There are some w ell-kno wn NP -hard optimisation problems that ha v e the highly desirable prop ert y of admitting a PT AS: examples inlude Knapsa k [32 ℄, Eulidean Tsp [2℄, and Independent Set restrited to planar graphs [6, 45 ℄. It is also w ell-kno wn that a large n um b er of optimisation problems do not admit PT AS unless some unexp eted ollapse of omplexit y lasses o urs. F or instane, problems lik e Max k -SA T [4℄ and Independent Set [5℄ do not admit a PT AS unless P = NP . W e note that if Π is a problem that do es not admit a PT AS, then there exists a onstan t c > 1 su h that Π annot b e appro ximated within c in p olynomial time. The  onstr aint satisfation pr oblem ( CSP ) [52 ℄ and its optimisation v arian ts ha v e pla y ed an imp ortan t role in resear h on appro ximabilit y . F or example, it is w ell kno wn that the famous PCP theorem has an equiv alen t reform ulation in terms of inappro ximabilit y of some CSP [4, 25 , 55 ℄, and the reen t om binatorial pro of of this theorem [ 25 ℄ deals en tirely with CSPs. Other imp ortan t examples inlude Håstad's rst optimal inappro ximabilit y results [31 ℄ and the w ork around the unique games onjeture of Khot [15 , 39 , 40 ℄. W e will fo us on a lass of optimisation problems kno wn as the maximum  onstr aint satis- fation pr oblem ( Max CSP ). The most w ell-kno wn examples in this lass probably are Max k -SA T and Max Cut . W e are no w ready to formally dene our problem. Let D b e a nite set. A subset R ⊆ D n is a r elation and n is the arity of R . Let R ( k ) D b e the set of all k -ary relations on D and let R D = ∪ ∞ i =1 R ( i ) D . A  onstr aint language is a nite subset of R D . Denition 1.1 ( CSP (Γ) ) The  onstr aint satisfation pr oblem over the  onstr aint language Γ , denote d CSP (Γ) , is dene d to b e the de ision pr oblem with instan e ( V , C ) , wher e • V is a set of variables, and • C is a  ol le tion of  onstr aints { C 1 , . . . , C q } , in whih e ah  onstr aint C i is a p air ( R i , s i ) with s i a list of variables of length n i ,  al le d the  onstr aint s op e, and R i ∈ Γ is an n i -ary r elation in R D ,  al le d the  onstr aint r elation. The question is whether ther e exists an assignment s : V → D whih satises al l  on- str aints in C or not. A  onstr aint ( R i , ( v i 1 , v i 2 , . . . , v i n i )) ∈ C is satised by an assign- ment s if the image of the  onstr aint s op e is a memb er of the  onstr aint r elation, i.e., if ( s ( v i 1 ) , s ( v i 2 ) , . . . , s ( v i n i )) ∈ R i . Man y om binatorial problems are subsumed b y the CSP framew ork; examples inlude problems in graph theory [30 ℄, om binatorial optimisation [38 ℄, and omputational learn- ing [22 ℄. W e refer the reader to [17 ℄ for an in tro dution to this framew ork. F or a onstrain t language Γ ⊆ R D , the optimisation problem Max CSP (Γ) is dened as follo ws: Denition 1.2 ( Max CSP (Γ) ) Max CSP (Γ) is dene d to b e the optimisation pr oblem with Instane: A n instan e ( V , C ) of CSP (Γ) . Solution: A n assignment s : V → D to the variables. 2 Measure: Numb er of  onstr aints in C satise d by the assignment s . W e use  ol le tions of onstrain ts instead of just sets of onstrain ts as w e do not ha v e an y w eigh ts in our denition of Max CSP . Some of our redutions will mak e use of opies of one onstrain t to sim ulate something whi h resem bles w eigh ts. W e  ho ose to use olletions instead of w eigh ts b eause b ounded o urrene restritions are easier to explain in the olletion setting. Note that w e pro v e our hardness results in this restrited setting without w eigh ts and with a onstan t b ound on the n um b er of o urrenes of ea h v ariable. Throughout this rep ort, Max CSP (Γ) - k will denote the problem Max CSP (Γ) restrited to instanes with the n um b er of o urrenes of ea h v ariable is b ounded b y k . F or our hardness results w e will write that Max CSP (Γ) - B is hard (in some sense) to denote that there is a k su h that Max CSP (Γ) - k is hard in this sense. If a v ariable o urs t times in a onstrain t whi h app ears s times in an instane, then this w ould on tribute t · s to the n um b er of o urrenes of that v ariable in the instane. Example: Giv en a (m ulti)graph G = ( V , E ) , the Max k -Cut problem, k ≥ 2 , is the problem of maximising | E ′ | , E ′ ⊆ E , su h that the subgraph G ′ = ( V , E ′ ) is k -olourable. F or k = 2 , this problem is kno wn simply as Max Cut . The problem Max k -Cut is kno wn to b e APX -omplete for an y k (it is Problem GT33 in [6 ℄), and so has no PT AS. Let N k denote the binary disequalit y relation on { 0 , 1 , . . . , k − 1 } , k ≥ 2 , that is, ( x, y ) ∈ N k ⇐ ⇒ x 6 = y . T o see that Max CSP ( { N k } ) is preisely Max k -Cut , think of v erties of a giv en graph as of v ariables, and apply the relation to ev ery pair of v ariables x, y su h that ( x, y ) is an edge in the graph, with the orresp onding m ultipliit y . Most of the early results on the omplexit y and appro ximabilit y of CSP and Max CSP w ere restrited to the Bo olean ase, i.e. when D = { 0 , 1 } . F or instane, S haefer [ 53 ℄  har- aterised the omplexit y of CSP (Γ) for all Γ o v er the Bo olean domain, the appro ximabilit y of Max CSP (Γ) for all Γ o v er the Bo olean domain ha v e also b een determined [19 , 20 , 38 ℄. It has b een noted that the study of non-Bo olean CSP seems to giv e a b etter understanding (when ompared with Bo olean CSP ) of what mak es CSP easy or hard: it app ears that man y observ ations made on Bo olean CSP are sp eial ases of more general phenomena. Reen tly , there has b een some ma jor progress in the understanding of non-Bo olean CSP : Bulato v has pro vided a omplete omplexit y lassiation of the CSP problem o v er a three-elemen t domain [9℄ and also giv en a lassiation of onstrain t languages that on tain all unary relations [ 7℄. Corresp onding results for Max CSP ha v e b een obtained b y Jonsson et al. [36 ℄ and Deinek o et al. [23 ℄. W e on tin ue this line of resear h b y studying t w o asp ets of non-Bo olean Max CSP . The omplexit y of CSP (Γ) is not kno wn for all onstrain t languages Γ  it is in fat a ma jor op en question [12 , 28 ℄. Ho w ev er, the piture is not ompletely unkno wn sine the omplexit y of CSP (Γ) has b een settled for man y onstrain t languages [ 9, 10 , 12 , 13 , 33 , 34 ℄. It has b een onjetured [28 ℄ that for all onstrain t languages Γ , CSP (Γ) is either in P or is NP -omplete, and the rened onjeture [12 ℄ (whi h w e refer to as the  CSP Conjeture, see Setion 3.2 for details) also desrib es the dividing line b et w een the t w o ases. Reall that if P 6 = NP , then Ladner's Theorem [43 ℄ states that there are problems of in termediate omplexit y , i.e., there are problems that are not in P and not NP -omplete. Hene, w e annot rule out a priori if there is a onstrain t language Γ su h that CSP (Γ) is neither in P nor NP -omplete. If the CSP Conjeture is true, then the family of onstrain t languages whi h are urren tly kno wn to mak e CSP (Γ) NP -omplete onsists of all onstrain t languages with this prop ert y . In the rst part of the rep ort w e study the family of all onstrain t languages Γ su h that it is urren tly kno wn that CSP (Γ) is NP -hard. W e pro v e that ea h onstrain t language in this family mak es Max CSP (Γ) ha v e a hard gap at lo ation 1. Hard gap at lo ation 1 means that 3 it is NP -hard to distinguish instanes of Max CSP (Γ) in whi h all onstrain ts are satisable from instanes where at most an ε -fration of the onstrain ts are satisable (for some onstan t ε whi h dep ends on Γ ) ¶ . It is immediate that ha ving a hard gap at lo ation 1 exludes the existene of a PT AS for the problem. The result is pro v ed in Setion 3 (Theorem 3.6 ) and an b e stated as follo ws (w e refer the reader to Setion 3 for an in tro dution to the algebrai terminology). Result A (Hardness at gap lo ation 1 for Max CSP ): Let Γ b e a ore onstrain t language and let A b e the algebra asso iated with Γ . If A c has a fator whi h only has pro jetions as term op erations, then Max CSP (Γ) has a hard gap at lo ation 1. The result holds ev en if w e ha v e a onstan t b ound on the n um b er of v ariable o urrenes. A similar result holds when the problem is restrited to satisable instanes only (Corol- lary 3.14 ). W e note that for the Bo olean domain and without the b ounded o urrene restri- tion, Result A follo ws from a result of Khanna et al. [38 , Theorem 5.14℄. In terestingly , the PCP theorem is equiv alen t to the fat that, for some onstrain t language Γ o v er some nite set D , Max CSP (Γ) has a hard gap at lo ation 1 [4 , 25 , 55 ℄. Clearly , CSP (Γ) annot b e p olynomial time solv able in this ase. F or an y onstrain t language Γ satisfying the ondition from Result A , the problem CSP (Γ) is kno wn to b e NP -omplete, and it has b een onjetured [12 ℄ that CSP (Γ) is in P for all other (ore) onstrain t languages (see Setion 3.2 ). Th us, Result A states that Max CSP (Γ) has a hard gap at lo ation 1 for any onstrain t language su h that CSP (Γ) is kno wn to b e NP -omplete. Moreo v er, if the ab o v e men tioned onjeture holds, then Max CSP (Γ) has a hard gap at lo ation 1 whenev er CSP (Γ) is not in P . Another equiv alen t reform ulation of the PCP theorem states that the problem Max 3-SA T has a hard gap at lo ation 1 [ 4 , 55 ℄, and our pro of of onsists of a gap preserving redution from this problem. W e also sho w ho w Result A an b e used for partially answ ering t w o op en questions. The rst one w as p osed b y Engebretsen et al. [ 26 ℄ and onerns the appro ximabilit y of a nite group problem while the seond w as p osed b y F eder et al. [ 27 ℄ and onerns the hardness of CSP (Γ) with the restrition that ea h v ariable o urs at most a onstan t n um b er of times, under the assumption that CSP (Γ) is NP -omplete. The te hniques w e use to pro v e Result A are partly from univ ersal algebra  su h metho ds ha v e previously pro v ed to b e v ery useful when studying the omplexit y of CSP [9 , 10 , 12 , 13 , 33 , 34 ℄. Ho w ev er, they ha v e not previously b een used to pro v e hardness results for Max CSP . T ypially , the algebrai om binatorial prop ert y of sup ermo dularit y and the te hnique of strit implemen tations (see Setion 4.3 and Setion 2.2 , resp etiv ely , for the denitions) ha v e b een used when pro ving results of this kind. This is, for instane, the ase in [23 , 36 ℄ where it is pro v ed that for an y onstrain t language Γ o v er D su h that Γ inludes the set C D = {{ ( x ) } | x ∈ D } or D has at most three elemen ts, Max CSP (Γ) is either solv able (to optimalit y) in p olynomial time or else APX -hard (in whi h ase it annot ha v e a PT AS unless P = NP ). The seond asp et of Max CSP w e study is the ase when the onstrain t language onsists of a single relation; this lass of problems on tains some of the most w ell-studied examples of Max CSP su h as Max Cut and Max DiCut . Before w e state this result w e need to dene some terminology . F or a relation R w e shall sa y that R is d -valid if ( d, . . . , d ) ∈ R for d ∈ D and simply valid if R is d -v alid for some d ∈ D . Informally , our main result on this problem is (see Theorem 4.1 for the formal statemen t): ¶ Some authors onsider the promise problem Gap-CSP [ ε, 1] where an instane is a Max CSP instane ( V , C ) and the problem is to deide b et w een the follo wing t w o p ossibilities: the instane is satisable, or at most ε · | C | onstrain ts are sim ultaneously satisable. Ob viously , if a Max CSP (Γ) has a hard gap at lo ation 1, then there exists an ε su h that the orresp onding Gap-CSP [ ε, 1] problem is NP -hard. 4 Result B (Appro ximabilit y of single relation Max CSP ): Let R b e a relation in R D . If R is empt y or v alid, then Max CSP ( { R } ) is trivial. Otherwise, there exists a onstan t c (whi h dep ends on R ) su h that it is NP -hard to appro ximate Max CSP ( { R } ) within c . The result holds ev en if w e ha v e a onstan t b ound on the n um b er of v ariable o urrenes. Jonsson and Krokhin [37 ℄ ha v e pro v ed that ev ery problem Max CSP ( { R } ) with R neither empt y nor v alid is NP -hard. W e strengthen their theorem b y pro ving Result B ; to do so, w e mak e use of Result A . Note that for some Max CSP problems su h appro ximation hardness results are kno wn, e.g., Max Cut and Max DiCut . Our result extends those hardness results to all p ossible relations. As a orollary to this result w e get that if Max CSP ( { R } ) is NP -hard, then there is no PT AS for Max CSP ( { R } ) (assuming P 6 = NP ). Note that a full omplexit y lassiation of single-relation CSP is not kno wn. In fat, F eder and V ardi [ 28 ℄ ha v e pro v ed that b y pro viding su h a lassiation, one has also lassied the CSP problem for al l onstrain t languages. In Setion 4.3 w e strengthen t w o earlier published results on Max CSP in v arious w a ys  the ommon theme is that Result B is used to obtain the results. The reader is referred to Setion 4.3 for denitions of the relev an t onepts used b elo w to desrib e the results. W e pro v e that unless P = NP , onstrain t languages whi h on tain all at most binary 2-monotone relations on a partially ordered set whi h is not a lattie order giv e rise to a Max CSP problem whi h is hard to appro ximate. The other result states that onstrain t languages whi h on tain all at most binary 2-monotone relations on a lattie and is not sup ermo dular on the lattie mak e Max CSP hard to appro ximate. These t w o problems ha v e previously b een studied b y Krokhin and Larose [41 , 42 ℄. Here is an o v erview of the rep ort: In Setion 2 w e dene some onepts w e need. Setion 3 on tains the pro of for our rst result and Setion 4 on tains the pro of of our seond result. In Setion 4.3 w e strengthen some earlier published results on Max CSP as men tioned ab o v e. W e giv e a few onluding remarks in Setion 5 . 2 Preliminaries A  ombinatorial optimisation pr oblem is dened o v er a set of instan es (admissible input data); ea h instane I has a set sol ( I ) of fe asible solutions asso iated with it, and ea h solution y ∈ sol ( I ) has a v alue m ( I , y ) . The ob jetiv e is, giv en an instane I , to nd a feasible solution of optim um v alue. The optimal v alue is the largest one for maximisation problems and the smallest one for minimisation problems. A om binatorial optimisation problem is said to b e an NP optimisation ( NPO ) problem if its instanes and solutions an b e reognised in p olynomial time, the solutions are p olynomially-b ounded in the input size, and the ob jetiv e funtion an b e omputed in p olynomial time (see, e.g., [6℄). Denition 2.1 (P erformane ratio) A solution s ∈ so l ( I ) to an instan e I of an NPO pr oblem Π is r -appr oximate if max  m ( I , s ) opt ( I ) , opt ( I ) m ( I , s )  ≤ r, wher e opt ( I ) is the optimal value for a solution to I . A n appr oximation algorithm for an NPO pr oblem Π has p erformane ratio R ( n ) if, given any instan e I of Π with |I | = n , it outputs an R ( n ) -appr oximate solution. PO is the lass of NPO problems that an b e solv ed (to optimalit y) in p olynomial time. An NPO problem Π is in the lass APX if there is a p olynomial time appro ximation algorithm 5 for Π whose p erformane ratio is b ounded b y a onstan t. The follo wing result is w ell-kno wn (see, e.g., [16 , Prop osition 2.3℄). Lemma 2.2 L et D b e a nite set. F or every  onstr aint language Γ ⊆ R D , Max CSP (Γ) b elongs to APX . Mor e over, if a is the maximum arity of any r elation in Γ , then ther e is a p olynomial time algorithm whih, for every instan e I = ( V , C ) of Max CSP (Γ) , pr o du es a solution satisfying at le ast | C | | D | a  onstr aints. Denition 2.3 (Hard to appro ximate) W e say that a pr oblem Π is hard to appro ximate if ther e exists a  onstant c suh that, Π is NP -har d to appr oximate within c (that is, the existen e of a p olynomial-time appr oximation algorithm for Π with p erforman e r atio c implies P = NP ). The follo wing notion has b een dened in a more general setting b y P etrank [50 ℄. Denition 2.4 (Hard gap at lo ation α ) Max CSP (Γ) has a hard gap at lo ation α ≤ 1 if ther e exists a  onstant ε < α and a p olynomial-time r e dution fr om an NP - omplete pr oblem Π to Max CSP (Γ) suh that, • Yes instan es of Π ar e mapp e d to instan es I = ( V , C ) suh that opt ( I ) ≥ α | C | , and • No instan es of Π ar e mapp e d to instan es I = ( V , C ) suh that opt ( I ) ≤ ε | C | . Note that if a problem Π has a hard gap at lo ation α (for an y α ) then Π is hard to appro ximate. This simple observ ation has b een used to pro v e inappro ximabilit y results for a large n um b er of optimisation problems. See, e.g., [ 3, 6 , 55 ℄ for surv eys on inappro ximabilit y results and the related PCP theory . 2.1 Appro ximation Preserving Redutions T o pro v e our appro ximation hardness results w e use AP -r e dutions . This t yp e of redution is most ommonly used to dene ompleteness for ertain lasses of optimisation problems (i.e., APX ). Ho w ev er, no APX -hardness results are atually pro v en in this rep ort sine w e onen trate on pro ving that problems are hard to appro ximate (in the sense of Denition 2.3 ). W e will frequen tly use AP -redutions an yw a y and this is justied b y Lemma 2.6 b elo w. Our denition of AP -redutions follo ws [20 , 38 ℄. Denition 2.5 ( AP -redution) Given two NPO pr oblems Π 1 and Π 2 an AP -redution fr om Π 1 to Π 2 is a triple ( F, G, α ) suh that, • F and G ar e p olynomial-time  omputable funtions and α is a  onstant; • for any instan e I of Π 1 , F ( I ) is an instan e of Π 2 ; • for any instan e I of Π 1 , and any fe asible solution s ′ of F ( I ) , G ( I , s ′ ) is a fe asible solution of I ; • for any instan e I of Π 1 , and any r ≥ 1 , if s ′ is an r -appr oximate solution of F ( I ) then G ( I , s ′ ) is an (1 + ( r − 1) α + o (1)) -appr oximate solution of I wher e the o -notation is with r esp e t to |I | . If suh a triple exist we say that Π 1 is AP -r e duible to Π 2 . W e use the notation Π 1 ≤ AP Π 2 to denote this fat. 6 It is a w ell-kno wn fat (see, e.g., Setion 8.2.1 in [6 ℄) that AP -redutions omp ose. The follo wing simple lemma mak es AP -redutions useful to us. Lemma 2.6 If Π 1 ≤ AP Π 2 and Π 1 is har d to appr oximate, then Π 2 is har d to appr oximate. Pro of: Let c > 1 b e the onstan t su h that it is NP -hard to appro ximate Π 1 within c . Let ( F, G, α ) b e the AP -redution whi h redues Π 1 to Π 2 . W e will pro v e that it is NP -hard to appro ximate Π 2 within r = 1 α ( c − 1) + 1 − ε ′ for an y ε ′ > 0 . Let I 1 b e an instane of Π 1 . Then, I 2 = F ( I 1 ) is an instane of Π 2 . Giv en an r -appro ximate solution to I 2 w e an onstrut an (1 + ( r − 1 ) α + o (1)) -appro ximate solution to I 1 using G . Hene, w e get an 1 + ( r − 1) α + o (1) = c − αε ′ + o (1) appro ximate solution to I 1 , and when the instanes are large enough this is stritly smaller than c . As c > 1 w e an  ho ose ε ′ su h that ε ′ > 0 and c − αε ′ > 1 . ✷ 2.2 Redution T e hniques The basi redution te hnique in our appro ximation hardness pro ofs is based on strit im- plementations and p erfe t implementations . Those te hniques ha v e b een used b efore when studying Max CSP and other CSP -related problems [20 , 36 , 38℄. Denition 2.7 (Implemen tation) A  ol le tion of  onstr aints C 1 , . . . , C m over a tuple of variables x = ( x 1 , . . . , x p )  al le d primary v ariables and y = ( y 1 , . . . , y q )  al le d auxiliary v ari- ables is an α -implemen tation of the p -ary r elation R for a p ositive inte ger α if the fol lowing  onditions ar e satise d: 1. F or any assignment to x and y , at most α  onstr aints fr om C 1 , . . . , C m ar e satise d. 2. F or any x suh that x ∈ R , ther e exists an assignment to y suh that exatly α  onstr aints ar e satise d. 3. F or any x , y suh that x 6∈ R , at most ( α − 1)  onstr aints ar e satise d. Denition 2.8 (Strit/P erfet Implemen tation) A n α -implementation is a strit imple- men tation if for every x suh that x 6∈ R ther e exists y suh that exatly ( α − 1)  onstr aints ar e satise d. A n α -implementation (not ne  essarily strit) is a p erfet implemen tation if α = m . It will sometimes b e on v enien t for us to view relations as prediates instead. In this ase an n -ary relation R o v er the domain D is a funtion r : D n → { 0 , 1 } su h that r ( x ) = 1 ⇐ ⇒ x ∈ R . Most of the time w e will use prediates when w e are dealing with strit implemen tations and relations when w e are w orking with p erfet implemen tations, b eause p erfet implemen tations are naturally written as a onjuntion of onstrain ts whereas strit implemen tations ma y naturally b e seen as a sum of prediates. W e will write strit α -implemen tations in the follo wing form g ( x ) + ( α − 1 ) = max y m X i =1 g i ( x i ) where x = ( x 1 , . . . , x p ) are the primary v ariables, y = ( y 1 , . . . , y q ) are the auxiliary v ariables, g ( x ) is the prediate whi h is implemen ted, and ea h x i is a tuple of v ariables from x and y . 7 W e sa y that a olletion of relations Γ stritly (p erfe tly) implements a relation R if, for some α ∈ Z + , there exists a strit (p erfet) α -implemen tation of R using relations only from Γ . It is not diult to sho w that if R an b e obtained from Γ b y a series of strit (p erfet) implemen tations, then it an also b e obtained b y a single strit (p erfet) implemen tation (for the Bo olean ase, this is sho wn in [ 20 , Lemma 5.8℄). The follo wing lemma indiates the imp ortane of strit implemen tations for Max CSP . It w as rst pro v ed for the Bo olean ase, but without the assumption on b ounded o urrenes, in [20 , Lemma 5.17℄. A pro of of this lemma in our setting an b e found in [23 , Lemma 3.4℄ (the lemma is stated in a sligh tly dieren t form but the pro of establishes the required AP - redution). Lemma 2.9 If Γ stritly implements a pr e di ate f , then, for any inte ger k , ther e is an inte ger k ′ suh that Max CSP (Γ ∪ { f } ) - k ≤ AP Max CSP (Γ) - k ′ . Lemma 2.9 will b e used as follo ws in our pro ofs of appro ximation hardness: if Γ ′ is a xed nite olletion of prediates ea h of whi h an b e stritly implemen ted b y Γ , then w e an assume that Γ ′ ⊆ Γ . F or example, if Γ on tains a binary prediate f , then w e an assume, at an y time when it is on v enien t, that Γ also on tains f ′ ( x, y ) = f ( y, x ) , sine this equalit y is a strit 1-implemen tation of f ′ . F or pro ving hardness at gap lo ation 1, w e ha v e the follo wing lemma. Lemma 2.10 If a nite  onstr aint language Γ p erfe tly implements a r elation R and Max CSP (Γ ∪ { R } ) - k has a har d gap at lo  ation 1, then Max CSP (Γ) - k ′ has a har d gap at lo  ation 1 for some inte ger k ′ . Pro of: Let N b e the minim um n um b er of relations that are needed in a p erfet implemen- tation of R using relations from Γ . Giv en an instane I = ( V , C ) of Max CSP (Γ ∪ { R } ) - k , w e onstrut an instane I ′ = ( V ′ , C ′ ) of Max CSP (Γ) - k ′ (where k ′ will b e sp eied b elo w) as follo ws: w e use the set V ′′ to store auxiliary v ariables during the redution so w e initially let V ′′ b e the empt y set. F or a onstrain t c = ( Q, s ) ∈ C , there are t w o ases to onsider: 1. If Q 6 = R , then add N opies of c to C ′ . 2. If Q = R , then add the implemen tation of R to C ′ where an y auxiliary v ariables in the implemen tation are replaed with fresh v ariables whi h are added to V ′′ . Finally , let V ′ = V ∪ V ′′ . It is lear that there exists an in teger k ′ , indep enden t of I , su h that I ′ is an instane of Max CSP (Γ ′ ) - k ′ . If all onstrain ts are sim ultaneously satisable in I , then all onstrain ts in I ′ are also sim ultaneously satisable. On the other hand, if opt ( I ) ≤ ε | C | then opt ( I ′ ) ≤ εN | C | + (1 − ε )( N − 1) | C | = ( ε + (1 − ε )(1 − 1 / N )) | C ′ | . The inequalit y holds b eause ea h onstrain t in I in tro dues a group of N onstrain ts in I ′ and, as opt ( I ) ≤ ε | C | , at most ε | C | su h groups are ompletely satised. In all other groups (there are (1 − ε ) | C | su h groups) at least one onstrain t is not satised. W e onlude that Max CSP (Γ) - k ′ has a hard gap at lo ation 1. ✷ An imp ortan t onept is that of a  or e . T o dene ores formally w e need retrations. A r etr ation of a onstrain t language Γ ⊆ R D is a funtion π : D → D su h that if D ′ is the image 8 of π then π ( x ) = x for all x ∈ D ′ , furthermore for ev ery R ∈ Γ w e ha v e ( π ( t 1 ) , . . . , π ( t n )) ∈ R for all ( t 1 , . . . , t n ) ∈ R . W e will sa y that Γ is a  or e if the only retration of Γ is the iden tit y funtion. Giv en a relation R ∈ R ( k ) D and a subset X of D w e dene the r estrition of R onto X as follo ws: R   X = { x ∈ X k | x ∈ R } . F or a set of relations Γ w e dene Γ   X = { R   X | R ∈ Γ } . If π is a retration of Γ with image D ′ ,  hosen su h that | D ′ | is minimal, then a ore of Γ is the set Γ   D ′ . F or onstrain t language Γ , Γ ′ w e sa y that Γ r etr ats to Γ ′ if there is a retration π of Γ su h that π (Γ) = Γ ′ . The in tuition here is that if Γ is not a ore, then it has a non-injetiv e retration π , whi h implies that, for ev ery assignmen t s , there is another assignmen t π s that satises all onstrain ts satised b y s and uses only a restrited set of v alues. Consequen tly the problem is equiv alen t to a problem o v er this smaller set. As in the ase of graphs, all ores of Γ are isomorphi, so one an sp eak ab out the ore of Γ . Example: Ev ery onstrain t language Γ on taining all unary relations is a ore b eause the only retration of the set of unary relations is the iden tit y op eration. The follo wing simple lemma onnets ores with non-appro ximabilit y . Lemma 2.11 If Γ ′ is the  or e of Γ , then, for any k , Max CSP (Γ ′ ) - k has a har d gap at lo  ation 1 if and only if Max CSP (Γ) - k has a har d gap at lo  ation 1. Pro of: Let π b e the retration of Γ su h that Γ ′ = { π ( R ) | R ∈ Γ } , where π ( R ) = { π ( t ) | t ∈ R } . Giv en an instane I = ( V , C ) of Max CSP (Γ) - k , w e onstrut an instane I ′ = ( V , C ′ ) of Max CSP (Γ ′ ) - k b y replaing ea h onstrain t ( R, s ) ∈ C b y ( π ( R ) , s ) . F rom a solution s to I ′ , w e onstrut a solution s ′ to I ′ su h that s ′ ( x ) = π ( s ( x )) . Let ( R, s ) ∈ C b e a onstrain t whi h is satised b y s . Then, there is a tuple x ∈ R su h that s ( s ) = x so π ( x ) ∈ π ( R ) and s ′ ( s ) = π ( s ( s )) = π ( x ) ∈ π ( R ) . Con v ersely , if ( π ( R ) , s ) is a onstrain t in I ′ whi h is satised b y s ′ , then there is a tuple x ∈ R su h that s ′ ( s ) = π ( s ( s )) = π ( x ) ∈ π ( R ) , and s ( s ) = x ∈ R . W e onlude that m ( I , s ) = m ( I ′ , s ′ ) . It is not hard to see that w e an do this redution in the other w a y to o, i.e., giv en an instane I ′ = ( V ′ , C ′ ) of Max CSP (Γ ′ ) - k , w e onstrut an instane I of Max CSP (Γ) - k b y replaing ea h onstrain t ( π ( R ) , s ) ∈ C ′ b y ( R, s ) . By the same argumen t as ab o v e, this diretion of the equiv alene follo ws, and w e onlude that the lemma is v alid. ✷ An analogous result holds for the CSP problem, i.e., if Γ ′ is the ore of Γ , then CSP (Γ) is in P ( NP -omplete) if and only if CSP (Γ ′ ) is in P ( NP -omplete); see [ 33 ℄ for a pro of. Cores pla y an imp ortan t role in Setion 4, to o. W e ha v e the follo wing lemma: Lemma 2.12 (Lemma 2.11 in [36 ℄) If Γ ′ is the  or e of Γ , then Max CSP (Γ ′ ) - B ≤ AP Max CSP (Γ) - B . The lemma is stated in a sligh tly dieren t form in [ 36 ℄ but the pro of establishes the required AP -redution. 3 Result A: Hardness at Gap Lo ation 1 for Max CSP In this setion, w e will pro v e Result A whi h w e state as Theorem 3.6 . The pro of mak es use of some onepts from univ ersal algebra and w e presen t the relev an t denitions and results in Setion 3.1 and Setion 3.2 . The pro of is on tained in Setion 3.3 . 9 3.1 Denitions and Results from Univ ersal Algebra W e will no w presen t the denitions and basi results w e need from univ ersal algebra. F or a more thorough treatmen t of univ ersal algebra in general w e refer the reader to [14 , 18 ℄. The artile [12 ℄ on tains a presen tation of the relationship b et w een univ ersal algebra and onstrain t satisfation problems. An op er ation on a nite set D is an arbitrary funtion f : D k → D . An y op eration on D an b e extended in a standard w a y to an op eration on tuples o v er D , as follo ws: let f b e a k -ary op eration on D . F or an y olletion of k n -tuples, t 1 , t 2 , . . . , t k ∈ D n , the n -tuple f ( t 1 , t 2 , . . . , t k ) is dened as follo ws: f ( t 1 , t 2 , . . . , t k ) = ( f ( t 1 [1] , t 2 [1] , . . . , t k [1]) , f ( t 1 [2] , t 2 [2] , . . . , t k [2]) , . . . , f ( t 1 [ n ] , t 2 [ n ] , . . . , t k [ n ])) , where t j [ i ] is the i -th omp onen t in tuple t j . If f ( d, d, . . . , d ) = d for all d ∈ D , then f is said to b e idemp otent . An op eration f : D k → D whi h satises f ( x 1 , x 2 , . . . , x k ) = x i , for some i , is alled a pr oje tion . Let R b e a relation in the onstrain t language Γ . If f is an op eration su h that for all t 1 , t 2 , . . . , t k ∈ R w e ha v e f ( t 1 , t 2 , . . . , t k ) ∈ R , then R is said to b e invariant (or, in other w ords, losed) under f . If all onstrain t relations in Γ are in v arian t under f , then Γ is said to b e in v arian t under f . An op eration f su h that Γ is in v arian t under f is alled a p olymorphism of Γ . The set of all p olymorphisms of Γ is denoted P ol (Γ) . Giv en a set of op erations F , the set of all relations that is in v arian t under all the op erations in F is denoted In v ( F ) . Example: Let D = { 0 , 1 , 2 } and let R b e the direted yle on D , i.e., R = { (0 , 1) , (1 , 2) , (2 , 0 ) } . One p olymorphism of R is the op eration f : { 0 , 1 , 2 } 3 → { 0 , 1 , 2 } dened as f ( x, y , z ) = x − y + z (mo d 3) . This an b e v eried b y onsidering all p ossible om binations of three tuples from R and ev aluating f omp onen t-wise. Let K b e the omplete graph on D . It is w ell kno wn and not hard to  he k that if w e view K as a binary relation, then all idemp oten t p olymorphisms of K are pro jetions. W e on tin ue b y dening a losure op erator h·i on sets of relations: for an y set Γ ⊆ R D , the set h Γ i onsists of all relations that an b e expressed using relations from Γ ∪ { E Q D } (where E Q D denotes the equalit y relation on D ), onjuntion, and existen tial quan tiation. Those are the relations denable b y primitive p ositive formulae (pp-form ulae). As an example of a pp-form ula onsider the relations A = { (0 , 0 ) , (0 , 1) , (1 , 0) } and B = { (1 , 0) , (0 , 1) , (1 , 1) } o v er the Bo olean domain { 0 , 1 } . With those t w o relations w e an onstrut I = { (0 , 0) , ( 0 , 1) , (1 , 1 ) } with the pp-form ula I ( x, y ) ⇐ ⇒ ∃ z : A ( x, z ) ∧ B ( z , y ) . Note that pp-form ulae and p erfet implemen tations from Denition 2.8 are the same onept. In tuitiv ely , onstrain ts using relations from h Γ i are exatly those whi h an b e sim ulated b y onstrain ts using relations from Γ in the CSP problem. Hene, for an y nite subset Γ ′ of h Γ i , CSP (Γ ′ ) is not harder than CSP (Γ) . That is, if CSP (Γ ′ ) is NP -omplete for some nite subset Γ ′ of h Γ i , then CSP (Γ) is NP -omplete. If CSP (Γ) is in P , then CSP (Γ ′ ) is in P for ev ery nite subset Γ ′ of h Γ i . W e refer the reader to [ 34 ℄ for a further disussion on this topi. The sets of relations of the form h Γ i are referred to as r elational lones , or  o-lones . An alternativ e  haraterisation of relational lones is giv en in the follo wing theorem. Theorem 3.1 ([51 ℄) • F or every set Γ ⊆ R D , h Γ i = In v ( P ol (Γ)) . 10 • If Γ ′ ⊆ h Γ i , then P ol (Γ) ⊆ P ol (Γ ′ ) . W e will no w dene nite algebras and some related notions whi h w e need later on. The three denitions b elo w losely follo w the presen tation in [12 ℄. Denition 3.2 (Finite algebra) A nite algebra is a p air A = ( A ; F ) wher e A is a nite non-empty set and F is a set of nitary op er ations on A . W e will only mak e use of nite algebras so w e will write algebr a instead of nite algebr a . An algebra is said to b e non-trivial if it has more than one elemen t. Denition 3.3 (Homomorphism of algebras) Given two algebr as A = ( A ; F A ) and B = ( B ; F B ) suh that F A = { f A i | i ∈ I } , F B = { f B i | i ∈ I } and b oth f A i and f B i ar e n i -ary for al l i ∈ I , then ϕ : A → B is said to b e an homomorphism fr om A to B if ϕ ( f A i ( a 1 , a 2 , . . . , a n i )) = f B i ( ϕ ( a 1 ) , ϕ ( a 2 ) , . . . , ϕ ( a n i )) for al l i ∈ I and a 1 , a 2 , . . . , a n i ∈ A . If ϕ is surje tive, then B is a homomorphi image of A . Giv en a homomorphism ϕ mapping A = ( A ; F A ) to B = ( B ; F B ) , w e an onstrut a equiv alene relation θ on A as θ = { ( x, y ) | ϕ ( x ) = ϕ ( y ) } . The relation θ is said to b e a  ongruen e r elation of A . W e an no w onstrut the quotient algebr a A /θ = ( A/θ ; F A /θ ) . Here, A/θ = { x/θ | x ∈ A } and x/θ is the equiv alene lass on taining x . F urthermore, F A /θ = { f /θ | f ∈ F A } and f / θ is dened su h that f / θ ( x 1 /θ , x 2 /θ , . . . , x n /θ ) = f ( x 1 , x 2 , . . . , x n ) /θ . F or an op eration f : D n → D and a subset X ⊆ D w e dene f   X as the funtion g : X n → D su h that g ( x ) = f ( x ) for all x ∈ X n . F or a set of op erations F on D w e dene F   X = { f   X | f ∈ F } . Denition 3.4 (Subalgebra) L et A = ( A ; F A ) b e an algebr a and B ⊆ A . If for e ah f ∈ F A and any b 1 , b 2 , . . . , b n ∈ B , we have f ( b 1 , b 2 , . . . , b n ) ∈ B , then B = ( B ; F A   B ) is a subalgebra of A . The op erations in P ol ( In v ( F A )) are the term op er ations of A . If all term op erations are surjetiv e, then the algebra is said to b e surje tive . Note that In v ( F A ) is a ore if and only if A is surjetiv e [12 , 33 ℄. If F onsist of all the idemp oten t term op erations of A , then the algebra ( A ; F ) is alled the ful l idemp otent r e dut of A , and w e will denote this algebra b y A c . Giv en a set of relations Γ o v er the domain D w e sa y that the algebra A Γ = ( D ; P ol (Γ)) is asso iate d with Γ . An algebra B is said to b e a fator of the algebra A if B is a homomorphi image of a subalgebra of A . A non-trivial fator is an algebra whi h is not trivial, i.e., it has at least t w o elemen ts. 3.2 Constrain t Satisfation and Algebra W e on tin ue b y desribing some onnetions b et w een onstrain t satisfation problems and univ ersal algebra. W e will also formally state Result A in Theorem 3.6 . The follo wing theorem onerns the hardness of CSP for ertain onstrain t languages. Theorem 3.5 ([12 ℄) L et Γ b e a  or e  onstr aint language. If A c Γ has a non-trivial fator whose term op er ations ar e only pr oje tions, then CSP (Γ) is NP -har d. It has b een onjetured [12 ℄ that, for all other ore languages Γ , the problem CSP (Γ) is tratable, and this onjeture has b een v eried in man y imp ortan t ases (see, e.g., [ 7 , 9℄). The rst main result of this rep ort is the follo wing theorem whi h states that Max CSP (Γ) - B has a hard gap at lo ation 1 whenev er the ondition whi h mak es CSP (Γ) hard in Theo- rem 3.5 is satised. 11 Theorem 3.6 L et Γ b e a  or e  onstr aint language. If A c Γ has a non-trivial fator whose term op er ations ar e only pr oje tions, then Max CSP (Γ) - B has a har d gap at lo  ation 1. The pro of of this result an b e found in Setion 3.3 . Note that if the ab o v e onjeture is true then Theorem 3.6 desrib es all onstrain t languages Γ for whi h Max CSP (Γ) has a hard gap at lo ation 1 b eause, ob viously , Γ annot ha v e this prop ert y when CSP (Γ) is tratable. There is another  haraterisation of the algebras in Theorem 3.5 whi h orresp onds to tratable onstrain t languages. T o state the  haraterisation w e need the follo wing denition. Denition 3.7 (W eak Near-Unanimit y F untion) A n op er ation f : D n → D , wher e n ≥ 2 , is a w eak near-unanimit y funtion if f is idemp otent and f ( x, y , y , . . . , y ) = f ( y , x, y , y , . . . , y ) = . . . = f ( y , . . . , y, x ) for al l x, y ∈ D . Hereafter w e will use the aron ym wnuf for w eak near-unanimit y funtions. W e sa y that an algebra A admits a wnuf if there is a wn uf among the term op erations of A . W e also sa y that a onstrain t language Γ admits a wn uf if there is a wn uf among the p olymorphisms of Γ . By om bining a theorem pro v en b y Maróti and MKenzie [ 48 , Theorem 1.1℄ with a result b y Bulato v and Jea v ons [ 11 , Prop osition 4.14℄, w e get the follo wing: Theorem 3.8 L et A b e an idemp otent algebr a. The fol lowing ar e e quivalent: • Ther e is a non-trivial fator B of A suh that B only have pr oje tions as term op er ations. • The algebr a A do es not admit any wnuf. 3.3 Pro of of Result A W e will no w pro v e Theorem 3.6 . Let 3 S AT 0 denote the relation { 0 , 1 } 3 \ { (0 , 0 , 0 ) } . W e also in tro due three sligh t v ariations of 3 S AT 0 , let 3 S AT 1 = { 0 , 1 } 3 \ { (1 , 0 , 0) } , 3 S AT 2 = { 0 , 1 } 3 \ { (1 , 1 , 0) } , and 3 S AT 3 = { 0 , 1 } 3 \ { (1 , 1 , 1) } . T o simplify the notation w e let Γ 3 S AT = { 3 S AT 0 , 3 S AT 1 , 3 S AT 2 , 3 S AT 3 } . It is not hard to see that the problem Max CSP (Γ 3 S AT ) is preisely Max 3Sa t . It is w ell-kno wn that this problem, ev en when restrited to instanes in whi h ea h v ariable o urs at most a onstan t n um b er of times, has a hard gap at lo ation 1, see e.g., [55 , Theorem 7℄. W e state this as a lemma. Lemma 3.9 ([55 ℄) Max CSP (Γ 3 S AT ) - B has a har d gap at lo  ation 1. T o pro v e Theorem 3.6 w e will utilise exp ander gr aphs . Denition 3.10 (Expander graph) A d -r e gular gr aph G is an expander graph if, for any S ⊆ V [ G ] , the numb er of e dges b etwe en S and V [ G ] \ S is at le ast min( | S | , | V [ G ] \ S | ) . Expander graphs are frequen tly used for pro ving prop erties of Max CSP , f. [21 , 49 ℄. T yp- ially , they are used for b ounding the n um b er of v ariable o urrenes. A onrete onstrution of expander graphs has b een pro vided b y Lub otzky et al. [ 46 ℄. Theorem 3.11 A p olynomial-time algorithm T and a xe d inte ger N exist suh that, for any k > N , T ( k ) pr o du es a 14-r e gular exp ander gr aph with k (1 + o (1)) verti es. 12 There are four basi ingredien ts in the pro of of Theorem 3.6 . The rst three are Lemma 2.10 , Lemma 3.9, and the use of expander graphs to b ound the n um b er of v ariable o urrenes. W e also use an alternativ e  haraterisation (Lemma 3.12 ) of onstrain t languages satisfying the onditions of the theorem. This is a sligh t mo diation of a part of the pro of of Prop osition 7.9 in [12 ℄. The impliation b elo w is in fat an equiv alene and w e refer the reader to [ 12 ℄ for the details. Giv en a funtion f : D → D , and a relation R ∈ R D , the ful l pr eimage of R under f , denoted b y f − 1 ( R ) , is the relation { x | f ( x ) ∈ R } (as usual, f ( x ) denotes that f should b e applied omp onen t wise to x ). Lemma 3.12 L et Γ b e a  or e  onstr aint language. If the algebr a A c Γ has a non-trivial fator whose term op er ations ar e only pr oje tions, then ther e is a subset B of D and a surje tive map- ping ϕ : B → { 0 , 1 } suh that the r elational lone h Γ ∪ C D i  ontains the r elations ϕ − 1 (3 S AT 0 ) , ϕ − 1 (3 S AT 1 ) , ϕ − 1 (3 S AT 2 ) , and ϕ − 1 (3 S AT 3 ) } . Pro of: Let A ′ b e the subalgebra of A su h that there is a homomorphism ϕ from A ′ to an algebra B whose term op erations are only pro jetions. W e an assume, without loss of generalit y , that the set { 0 , 1 } is on tained in the univ erse of B . It is easy to see that an y relation is in v arian t under an y pro jetions. Sine B only ha v e pro jetions as term op erations, the four relations 3 S AT 0 , 3 S AT 1 , 3 S AT 2 and 3 S AT 3 are in v arian t under the term op erations of B . It is not hard to  he k (see [ 12 ℄) that the full preimages of those relations under ϕ are in v arian t under the term op erations of A ′ and therefore they are also in v arian t under the term op erations of A . By Theorem 3.1 , this implies { ϕ − 1 (3 S AT 0 ) , ϕ − 1 (3 S AT 1 ) , ϕ − 1 (3 S AT 2 ) , ϕ − 1 (3 S AT 3 ) } ⊆ h Γ ∪ C D i . ✷ W e are no w ready to presen t the pro of of Theorem 3.6 . Let S b e a p erm utation group on the set X . An orbit of S is a subset Ω of X su h that Ω = { g ( x ) | g ∈ S } for some x ∈ X . Pro of: Let A Γ = ( D ; P ol (Γ)) b e the algebra asso iated with Γ . F or an y a ∈ D , w e denote the unary onstan t relation on taining only a b y c a , i.e., c a = { ( a ) } . Let C D denote the set of all onstan t relations o v er D , that is, C D = { c a | a ∈ D } . By Lemma 3.12 , there exists a subset (in fat, a subalgebra) B of D and a surjetiv e mapping ϕ : B → { 0 , 1 } su h that the relational lone h Γ ∪ C D i on tains ϕ − 1 (Γ 3 S AT ) = { ϕ − 1 ( R ) | R ∈ Γ 3 S AT } . By Lemma 3.9 , Max CSP (Γ 3 S AT ) - B is hard at gap lo ation 1, so, b y Lemma 2.11 , Max CSP ( ϕ − 1 (Γ 3 S AT )) - B is also hard at gap lo ation 1 (b eause Γ 3 S AT is the ore of ϕ − 1 (Γ 3 S AT ) ). Sine Γ is a ore, its unary p olymorphisms form a p erm utation group S on D . W e an without loss of generalit y assume that D = { 1 , . . . , p } . It is kno wn (see Prop osition 1.3 of [ 54 ℄) and not hard to  he k (using Theorem 3.1 ) that Γ an p erfetly implemen t the follo wing relation: R S = { ( g (1) , . . . , g ( p )) | g ∈ S } . Then it an also p erfetly implemen t the relations E Q i for 1 ≤ i ≤ p where E Q i is the restrition of the equalit y relation on D to the orbit in S whi h on tains i . W e ha v e E Q i ( x, y ) ⇐ ⇒ ∃ z 1 , . . . , z i − 1 , z i +1 , . . . , z p : R S ( z 1 , . . . , z i − 1 , x, z i +1 , . . . , z p ) ∧ R S ( z 1 , . . . , z i − 1 , y , z i +1 , . . . , z p ) . F or 0 ≤ i ≤ 3 , let R i b e the preimage of 3 S AT i under ϕ . Sine R i ∈ h Γ ∪ C D i , w e an sho w that there exists a ( p + 3) -ary relation R ′ i in h Γ i su h that R i = { ( x, y , z ) | (1 , 2 , . . . , p, x, y, z ) ∈ R ′ i } . Indeed, sine R i ∈ h Γ ∪ C D i , R i an b e dened b y a pp-form ula R i ( x, y , z ) ⇐ ⇒ ∃ t ψ ( t , x, y , z ) (here t denotes a tuple of v ariables) where ψ is a onjuntion of atomi form ulas in v olving prediates from Γ ∪ C D and v ariables from t and { x, y , z } . Note that, in ψ , no prediate from 13 C D is applied to one of { x, y , z } b eause these v ariables an tak e more than one v alue in R i . W e an without loss of generalit y assume that ev ery prediate from C D app ears in ψ exatly one. Indeed, if su h a prediate app ears more than one, then w e an iden tify all v ariables to whi h it is applied, and if it do es not app ear at all then w e an add a new v ariable to t and apply this prediate to it. No w assume without loss of generalit y that the prediate c i , 1 ≤ i ≤ p , is applied to the v ariable t i in ψ , and ψ = ψ 1 ∧ ψ 2 where ψ 1 = V p i =1 c i ( t i ) and ψ 2 on tains only prediates from Γ \ C D . Let t ′ b e the list of v ariables obtained from t b y remo ving t 1 , . . . , t p . It no w is easy to  he k that that the ( p + 3 ) -ary relation R ′ i dened b y the pp-form ula ∃ t ′ ψ 2 ( t , x, y , z ) has the required prop ert y . Cho ose R ′ i to b e the minimal relation in h Γ i su h that R i = { ( x, y , z ) | (1 , 2 , . . . , p, x, y, z ) ∈ R ′ i } . W e will sho w that R ′ i = { ( g (1) , g (2) , . . . , g ( p ) , g ( x ) , g ( y ) , g ( z )) | g ∈ S, ( x, y , z ) ∈ R i } . The set on the righ t-hand side of the ab o v e equalit y m ust b e on tained in R ′ i b eause R ′ i is in v arian t under all op erations in S . On the other hand, if a tuple b = ( b 1 , . . . , b p , d, e, f ) b elongs to R ′ i , then there is a p erm utation g ∈ S su h that ( b 1 , . . . , b p ) = ( g (1) , . . . , g ( p )) (otherwise, the in tersetion of this relation with R S × D 3 ∈ h Γ i w ould giv e a smaller relation with the required prop ert y). No w note that the tuple (1 , . . . , p, g − 1 ( d ) , g − 1 ( e ) , g − 1 ( f )) also b elongs to R ′ i implying, b y the  hoie of R ′ i , that ( g − 1 ( d ) , g − 1 ( e ) , g − 1 ( f )) ∈ R i . This ompletes the pro of and the relation R ′ i is as desrib ed ab o v e. T o simplify the notation, let Γ ′ = { R ′ i | 0 ≤ i ≤ 3 } ∪ { E Q 1 , . . . , E Q p } . By Lemma 2.10 , in order to pro v e the theorem, it sues to sho w that Max CSP (Γ ′ ) - B has a hard gap at lo ation 1. By Lemma 3.9 , there is an in teger K su h that Max CSP (Γ 3 S AT ) - K has a hard gap at lo ation 1. Cho ose K su h that K > 14 . By Lemma 2.11 , Max CSP ( ϕ − 1 (Γ 3 S AT )) - K has the same prop ert y . W e will no w AP -redue Max CSP ( ϕ − 1 (Γ 3 S AT )) - K to Max CSP (Γ ′ ) - B . T ak e an arbitrary instane I = ( V , C ) of Max CSP ( ϕ − 1 (Γ 3 S AT )) - K , and build an instane I ′ = ( V ′ , C ′ ) of Max CSP (Γ ′ ) as follo ws: in tro due new v ariables u 1 , . . . , u p , and replae ea h onstrain t R i ( x, y , z ) in I b y R ′ i ( u 1 , . . . , u p , x, y , z ) . Note that ev ery v ariable, exept the u i 's, in I ′ app ears at most K times. W e will no w use expander graphs to onstrut an instane I ′′ of Max CSP (Γ ′ ) with a onstan t b ound on the n um b er of o urrenes for ea h v ariables. Let q b e the n um b er of onstrain ts in I and let q ′ = max { N , q } , where N is the onstan t in Theorem 3.11 . Let G = ( W, E ) b e an expander graph (onstruted in p olynomial time b y the algorithm T ( q ′ ) in Theorem 3.11 ) su h that W = { w 1 , w 2 , . . . , w m } and m ≥ q . The expander graph T ( q ′ ) ha v e q ′ (1 + o (1)) v erties. Hene, there is a onstan t α su h that T ( q ′ ) has at most αq v erties. F or ea h 1 ≤ j ≤ p , w e in tro due m fresh v ariables w j 1 , w j 2 , . . . , w j m in to I ′′ . F or ea h edge { w i , w k } ∈ E and 1 ≤ j ≤ p , in tro due p opies of the onstrain t E Q j ( w j i , w j k ) in to C ′′ . Let C 1 , C 2 , . . . , C q b e an en umeration of the onstrain ts in C ′ . Replae u j b y w j i in C i for all 1 ≤ i ≤ q . Finally , let C ∗ b e the union of the (mo died) onstrain ts in C ′ and the equalit y onstrain ts in C ′′ . It is lear that ea h v ariable o urs in I ′′ at most K p times (reall that p = | D | is a onstan t). Clearly , a solution s to I satisfying all onstrain ts an b e extended to a solution to I ′′ , also satisfying all onstrain ts, b y setting s ( w j i ) = j for all 1 ≤ i ≤ m and all 1 ≤ j ≤ p . On the other hand, if m ( I , s ) ≤ ε | C | , then let s ′ b e an optimal solution to I ′′ . W e will pro v e that there is a onstan t ε ′ < 1 (whi h dep ends on ε but not on I ) su h that m ( I ′′ , s ′ ) ≤ ε ′ | C ∗ | . W e rst pro v e that, for ea h 1 ≤ j ≤ p , w e an assume that all v ariables in W j = { w j 1 , w j 2 , . . . , w j m } ha v e b een assigned the same v alue b y s ′ and that all onstrain ts in C ′′ are 14 satised b y s ′ . W e sho w that giv en a solution s ′ to I ′′ , w e an onstrut another solution s 2 su h that m ( I ′′ , s 2 ) ≥ m ( I ′′ , s ′ ) and s 2 satises all onstrain ts in C ′′ . Let a j b e the v alue that at least m/p of the v ariables in W j ha v e b een assigned b y s ′ . W e onstrut the solution s 2 as follo ws: s 2 ( w j i ) = a j for all i and j , and s 2 ( x ) = s ′ ( x ) for all other v ariables. If there is some j su h that X = { x ∈ W j | s ′ ( x ) 6 = a j } is non-empt y , then, sine G is an expander graph, there are at least p · min ( | X | , | W j \ X | ) onstrain ts in C ′′ whi h are not satised b y s ′ . Note that b y the  hoie of X , w e ha v e | W j \ X | ≥ m /p whi h implies p · min( | X | , | W j \ X | ) ≥ | X | . By  hanging the v alue of the v ariables in X , w e will mak e at most | X | non-equalit y onstrain ts in C ∗ unsatised b eause ea h of the v ariables in W j o urs in at most one non-equalit y onstrain t in C ∗ . In other w ords, when the v alue of the v ariables in X are  hanged w e gain at least | X | in the measure as some of the equalit y onstrain ts in C ′′ will b eome satised, furthermore w e lose at most | X | b y making at most | X | onstrain ts in C ∗ unsatised. W e onlude that m ( I ′ , s 2 ) ≥ m ( I ′ , s ′ ) . Th us, w e ma y assume that all equalit y onstrain ts in C ′′ are satised b y s ′ . Sine the expander graph G is 14-regular and has at most αq v erties, it has at most 14 2 αq edges. Hene, the n um b er of equalit y onstrain ts in C ′′ is at most 7 αq p , and | C ′′ | / | C ′ | ≤ 7 αp . W e an no w b ound m ( I ′′ , s 2 ) as follo ws: m ( I ′′ , s 2 ) ≤ opt ( I ′ ) + | C ′′ | ≤ ε | C ′ | + | C ′′ | | C ′ | + | C ′′ | ( | C ′ | + | C ′′ | ) ≤ ε + 7 αp 1 + 7 αp ( | C ′ | + | C ′′ | ) . Sine | C ∗ | = | C ′ | + | C ′′ | , it remains to set ε ′ = ε +7 αp 1+7 αp . ✷ W e nish this setion b y using Theorem 3.6 to answ er, at least partially , t w o op en questions. The rst one onerns the omplexit y of CSP (Γ) - B . In partiular, the follo wing onjeture has b een made b y F eder et al. [27 ℄. Conjeture 3.13 F or any xe d Γ suh that CSP (Γ) is NP - omplete ther e is an inte ger k suh that CSP (Γ) - k is NP - omplete. Under the assumption that the CSP onjeture (that all problems CSP (Γ) not o v ered b y Theorem 3.5 are tratable) holds, an armativ e answ er follo ws immediately from Theorem 3.6 . So for all onstrain t languages Γ su h that CSP (Γ) is urren tly kno wn to b e NP -omplete it is also the ase that CSP (Γ) - B is NP -omplete. The seond result onerns the appro ximabilit y of equations o v er non-ab elian groups. P e- trank [50 ℄ has noted that hardness at gap lo ation 1 implies the follo wing: supp ose that w e restrit ourselv es to instanes of Max CSP (Γ) su h that there exist solutions that satisfy all onstrain ts, i.e. w e onen trate on satisable instanes. Then, there exists a onstan t c (de- p ending on Γ ) su h that no p olynomial-time algorithm an appro ximate this problem within c (unless P = NP ). W e get the follo wing result for satisable instanes: Corollary 3.14 L et Γ b e a  or e  onstr aint language and let A b e the algebr a asso iate d with Γ . Assume ther e is a fator B of A c suh that B only have pr oje tions as term op er ations. Then, ther e exists a  onstant c suh that Max CSP (Γ) - B r estrite d to satisable instan es  annot b e appr oximate d within c in p olynomial time (unless P = NP ). W e will no w use this observ ation for studying a problem onerning groups. Let G = ( G, · ) denote a nite group with iden tit y elemen t 1 G . An e quation o v er a set of v ariables V is an expression of the form w 1 · . . . · w k = 1 G , where w i (for 1 ≤ i ≤ k ) is either a v ariable, an in v erted v ariable, or a group onstan t. Engebretsen et al. [26 ℄ ha v e studied the follo wing problem: 15 Denition 3.15 ( Eq G ) The  omputational pr oblem Eq G (wher e G is a nite gr oup) is dene d to b e the optimisation pr oblem with Instane: A set of variables V and a  ol le tion of e quations E over V . Solution: A n assignment s : V → G to the variables. Measure: Numb er of e quations in E whih ar e satise d by s . The problem Eq 1 G [3℄ is the same as Eq G exept for the additional restritions that ea h equation on tains exatly three v ariables and no equation on tains the same v ariable more than one. Their main result w as the follo wing inappro ximabilit y result: Theorem 3.16 (Theorem 1 in [26 ℄) F or any nite gr oup G and  onstant ε > 0 , it is NP - har d to appr oximate Eq 1 G [3℄ within | G | − ε . Engebretsen et al. left the appro ximabilit y of Eq 1 G [3℄ for satisable instanes as an op en question. W e will giv e a partial answ er to the appro ximabilit y of satisable instanes of Eq G . It is not hard to see that for an y in teger k , the equations with at most k v ariables o v er a nite group an b e view ed as a onstrain t language. F or a group G , w e denote the onstrain t language whi h orresp onds to equations with at most three v ariables b y Γ G . Hene, for an y nite group G , the problem Max CSP (Γ G ) is no harder than Eq G . Goldmann and Russell [29 ℄ ha v e sho wn that CSP (Γ G ) is NP -hard for ev ery nite non- ab elian group G . This result w as extended to more general algebras b y Larose and Zádori [44 ℄. They also sho w ed that for an y non-ab elian group G , the algebra A = ( G ; P ol (Γ G )) has a non- trivial fator B su h that B only ha v e pro jetions as term op erations. W e no w om bine Larose and Zádori's result with Theorem 3.6 : Corollary 3.17 F or any nite non-ab elian gr oup G , Eq G has a har d gap at lo  ation 1. Th us, there is a onstan t c su h that no p olynomial-time algorithm an appro ximate satis- able instanes of Eq G b etter than c , unless P = NP . There also exists a onstan t k (dep ending on the group G ) su h that the result holds for instanes with v ariable o urrene b ounded b y k . 4 Result B: Appro ximabilit y of Single Relation Max CSP In this setion, w e will pro v e the follo wing theorem: Theorem 4.1 L et R ∈ R ( n ) D b e non-empty. If ( d, . . . , d ) ∈ R for some d ∈ D , then Max CSP ( { R } ) is solvable in line ar time. Otherwise, Max CSP ( { R } ) - B is har d to appr oximate. The pro of mak es ruial use of Theorem 3.6 and it an b e divided in to a n um b er of steps: 1. Lemma 4.8 together with Lemma 4.7 pro v es that direted yles are hard to appro ximate (i.e., the theorem holds when R is the edge relation of a direted yle). 2. V ertex-transitiv e digraphs whi h are not direted yles are pro v ed to b e hard to ap- pro ximate in Lemma 4.6 . 3. Lemma 4.10 giv e appro ximation hardness for bipartite digraphs. 4. Lemma 4.15 redues the non-v ertex transitiv e ase to the v ertex-transitiv e ase. 16 5. Lemma 4.17 redues general relations to binary relations, i.e., to digraphs. 6. Finally , Theorem 4.1 is pro v ed b y assem bling the results from the previous setions. As indiated b y the list ab o v e the bulk of the w ork deals with binary relations. 4.1 Appro ximabilit y of Binary Relations In this setion, w e will pro v e that non-empt y non-v alid binary relations giv e rise to Max CSP problems whi h are hard to appro ximate. Subsetion 4.1.1 deals with binary (not neessarily symmetri) relations ha ving a transitiv e automorphism group, and Setion 4.1.2 deals with general binary relations. Sometimes it will b e on v enien t for us to view binary relations as digraphs. A digr aph is a pair ( V , E ) su h that V is a nite set and E ⊆ V × V . A gr aph is a digraph ( V , E ) su h that for ev ery pair ( x, y ) ∈ E w e also ha v e ( y , x ) ∈ E . Let R ∈ R D b e a binary relation. As R is binary it an b e view ed as a digraph G with v ertex set V [ G ] = D and edge set E [ G ] = R . W e will mix freely b et w een those t w o notations. F or example, w e will sometimes write ( x, y ) ∈ G with the in tended meaning ( x, y ) ∈ E [ G ] = R . Let G b e a digraph, R = E [ G ] , and let Aut ( G ) denote the automorphism group of G . If Aut ( G ) is transitiv e (i.e., on tains a single orbit), then w e sa y that G is vertex-tr ansitive . If D an b e partitioned in to t w o sets, A and B , su h that for an y x, y ∈ A (or x, y ∈ B ) w e ha v e ( x, y ) 6∈ R , then R (and G ) is bip artite . The dir e te d yle of length n is the digraph G with v ertex set V [ G ] = { 0 , 1 , . . . , n − 1 } and edge set E [ G ] = { ( x, x + 1 ) | x ∈ V [ G ] } , where the addition is mo dulo n . Analogously , the undir e te d yle of length n is the graph H with v ertex set V [ H ] = { 0 , 1 , . . . , n − 1 } and edge set E [ H ] = { ( x, x +1) | x ∈ V [ H ] }∪ { ( x +1 , x ) | x ∈ V [ H ] } (also in this ase the additions are mo dulo n ). The undireted path with t w o v erties will b e denoted b y P 2 . 4.1.1 V ertex-transitiv e Digraphs W e will no w ta kle non-bipartite v ertex-transitiv e digraphs and pro v e that they giv e rise to Max CSP problems whi h are hard at gap lo ation 1. T o do this, w e mak e use of the algebrai framew ork whi h w e used and dev elop ed in Setion 3. Reall that w e denote the unary onstan t relations o v er a domain D b y C D , i.e., C D = {{ ( x ) } | x ∈ D } . W e will need ertain hardness results in the forthoming pro ofs. Theorem 4.2 ([8 ℄) L et G b e an undir e te d  or e gr aph and let A G b e the algebr a asso iate d with G . If G is not bip artite, then ther e is a fator of A c G whih only have pr oje tions as term op er ations. Lemma 4.3 L et G b e a vertex-tr ansitive  or e digr aph suh that | V [ G ] | = 3 or | V [ G ] | = 4 . If G is not a dir e te d yle, then G do es not admit a wnuf. Pro of: Let v and u b e t w o v erties in a v ertex-transitiv e ore digraph. Note that the in- and out-degrees of u and v m ust oinide, and hene the in- and out-degrees of v m ust b e the same. Ha ving this in mind, it is easy to see that there are only t w o digraphs with three v erties satisfying the onditions in the lemma: the direted yle and the omplete graph on three v erties. Similarly , it is easy to see that there are three ore digraphs on four v erties whi h are v ertex-transitiv e: the direted yle, the omplete graph on four v erties and the digraph in Figure 1 . 17 • / / •   • O O        • o o @ @ @ @ @ @ @ Figure 1: The non-trivial ase in Lemma 4.3 F or omplete graphs, the results follo ws from Theorems 4.2 and 3.8 . Denote the di- graph in Figure 1 b y G and onsider the follo wing p erfet implemen tation (originally used b y MaGillivra y [47 , step 3 in Theorem 3.4℄). H ( x, y ) ⇐ ⇒ ∃ u, v : G ( x, u ) ∧ G ( u, v ) ∧ G ( v, u ) ∧ G ( v, y ) It is not hard to see that H is the omplete graph on four v erties. Sine H do es not admit an y wn uf, Theorem 3.1 implies that G do es not admit a wn uf either. ✷ Lemma 4.4 L et G and H b e two digr aphs suh that ther e is a r etr ation fr om G to H . If G admits a wnuf, then so do es H . Pro of: Let r b e a retration from G to H . If G admits a wn uf f ( x 1 , . . . , x n ) , then it is easy to  he k that H admits the wn uf r ( f ( x 1 , . . . , x n )) . ✷ Lemma 4.5 If G is a vertex-tr ansitive digr aph that do es not r etr at to a dir e te d yle, then G admits no wnuf. Pro of: F or the sak e of on tradition, let G b e a digraph with the minim um n um b er of v erties su h that G is v ertex-transitiv e, do es not retrat to a direted yle and G admits a wn uf. F urthermore, among all oun terexamples with | V [ G ] | v erties, let G b e the one with the maxim um n um b er of edges. It is w ell kno wn, and easy to sho w, that the ore of a v ertex-transitiv e digraph is also v ertex-transitiv e. If G is not a ore, then the ore of G is v ertex-transitiv e, admits a wn uf (Lemma 4.4 ), and do es not retrat to a direted yle. Hene, if G is not a ore, then the ore of G is a smaller oun terexample. W e an therefore, without loss of generalit y , assume that G is a ore. By Lemma 4.3 , w e an assume that | V [ G ] | > 4 . W e need the latter assumption b eause this is assumed in the pro of of Theorem 3.4 in [ 47 ℄, whi h w e use b elo w. F or a digraph H , let undir ( H ) b e the digraph indued b y the double edges of H . It is easy to see that undir ( H ) an b e p erfetly implemen ted from H as follo ws: undir ( H )( x, y ) ⇐ ⇒ H ( x, y ) ∧ H ( y , x ) . (1) The pro of of Theorem 3.4 in [47℄ sho ws that it is p ossible to p erfetly implemen t a digraph H with G and C V [ G ] su h that there is a retration r from H to a digraph H ′ whi h is v ertex-transitiv e and 1. undir ( H ′ ) is not bipartite and not v alid, or 2. | V [ H ′ ] | < | V [ G ] | and H ′ do es not retrat to a direted yle, or 3. | V [ H ′ ] | = | V [ G ] | and | E [ H ′ ] | > | E [ G ] | and H ′ do es not retrat to a direted yle. 18 In fat, the pro of in [47 ℄ uses onstrutions alled indiator and subindiator to obtain H from G , but these onstrutions are w ell kno wn to preisely orresp ond to ertain p erfet implemen tations (or pp-form ulas, see [8 ℄ for details). Note that, sine an y wn uf is idemp oten t, the onstrain t language { E [ G ] } ∪ C V [ G ] admits a wn uf. Then, b y Theorem 3.1 , the digraph H admits a wn uf. Lemma 4.4 applied to H sho ws that H ′ admits a wn uf as w ell. No w w e see that ases (2) and (3) are imp ossible, sine H ′ w ould on tradit the  hoie of G . Case (1) leads to a on tradition to o b eause, b y Lemmas 4.2 and 4.4 , the ore of undir ( H ′ ) , whi h is a non-bipartite undireted graph, w ould also admit a wn uf whi h is imp ossible b y Theorems 4.2 and 3.8 . ✷ Corollary 4.6 L et H b e a vertex-tr ansitive  or e digr aph whih is not valid and not a dir e te d yle. Then, Max CSP ( { H } ) - B has a har d gap at lo  ation 1. Pro of: Immediately follo ws from Lemma 4.5 , Theorem 3.8 , and Theorem 3.6 ✷ The next lemmas help to deal with the remaining v ertex-transitiv e graphs, i.e. those that retrat to a direted yle. Lemma 4.7 If G is the undir e te d p ath with two verti es P 2 , or an undir e te d yle C k , k > 2 , then Max CSP ( { G } ) - B is har d to appr oximate. Pro of: If G = P 2 , then the result follo ws from Example 1. If G = C k and k is ev en, then the ore of C k is isomorphi to P 2 and the result follo ws from Lemmas 2.12 , 2.6 om bined with Example 1 . F rom no w on, assume that G = C k , k is o dd, and k ≥ 3 . W e will sho w that w e an stritly implemen t N k , i.e., the inequalit y relation. W e use the follo wing strit implemen tation N k ( z 1 , z k − 1 ) + ( k − 3) = max z 2 ,z 3 ,...,z k − 2 C k ( z 1 , z 2 ) + C k ( z 2 , z 3 ) + . . . + C k ( z k − 3 , z k − 2 ) + C k ( z k − 2 , z k − 1 ) . It is not hard to see that if z 1 6 = z k − 1 , then all k − 2 onstrain ts on the righ t hand side an b e satised. If z 1 = z k − 1 , then k − 3 onstrain ts are satised b y the assignmen t z i = z 1 + i − 1 , for all i su h that 1 < i < k − 1 (the addition and subtration are mo dulo k ). F urthermore, no assignmen t an satisfy all onstrain ts. T o see this, note that su h an assignmen t w ould dene a path z 1 , z 2 , . . . , z k − 1 in C k with k − 2 edges and z 1 = z k − 1 . This is imp ossible sine k − 2 is o dd and k − 2 < k . The lemma no w follo ws from Lemmas 2.9 and 2.6 together with Example 1. ✷ Lemma 4.8 If G is a digr aph suh that ( x, y ) ∈ E [ G ] ⇒ ( y , x ) 6∈ E [ G ] , then Max CSP ( { H } ) - B ≤ AP Max CSP ( { G } ) - B , wher e H is the undir e te d gr aph obtaine d fr om G by r eplaing every e dge in G by two e dges in opp osing dir e tions in H . Pro of: H ( x, y ) + (1 − 1) = G ( x , y ) + G ( y , x ) is a strit implemen tation of H and the result follo ws from Lemma 2.9 . ✷ Lemma 4.9 If G is a non-empty non-valid vertex-tr ansitive digr aph, then Max CSP ( { G } ) - B is har d to appr oximate. Pro of: By Lemmas 2.12 and 2.6 , it is enough to onsider ores. F or direted yles, the results follo ws from Lemmas 4.7 and 4.8 , and, for all other digraphs, from Corollary 4.6 . ✷ 19 4.1.2 General Digraphs The main lemma of this setion is Lemma 4.16 whi h pro v es our result for general digraphs. W e b egin b y onsidering bipartite digraphs. Lemma 4.10 If G is a bip artite digr aph whih is neither empty nor valid, then Max CSP ( { G } ) - B is har d to appr oximate. Pro of: If there are t w o edges ( x, y ) , ( y , x ) ∈ E [ G ] , then the ore of G is isomorphi to P 2 and the result follo ws from Lemmas 2.6 and 2.12 together with Example 1. If no su h pair of edges exist, then Lemmas 2.6 and 4.8 redue this ase to the previous ase where there are t w o edges ( x, y ) , ( y , x ) ∈ E [ G ] . ✷ W e will use a te hnique kno wn as domain r estrition [ 23 ℄ in the sequel. F or a subset D ′ ⊆ D , let Γ   D ′ = { R   D ′ | R ∈ Γ and R   D ′ is non-empt y } . The follo wing lemma w as pro v ed in [23 , Lemma 3.5℄ (the lemma is stated in a sligh tly dieren t form there, but the pro of together with [6, Lemma 8.2℄ and Lemma 2.2 implies the existene of the required AP -redution). Lemma 4.11 L et D ′ ⊆ D and D ′ ∈ Γ , then Max CSP (Γ   D ′ ) - B ≤ AP Max CSP (Γ) - B . T ypially , w e will let D ′ b e an orbit in the automorphism group of a graph. W e are no w ready to presen t the three lemmas that are the building blo  ks of Lemma 4.15 . Let G b e a digraph. F or a set A ⊆ V [ G ] , w e dene A + = { j | ( i, j ) ∈ E [ G ] , i ∈ A } , and A − = { i | ( i, j ) ∈ E [ G ] , j ∈ A } . Lemma 4.12 If a  onstr aint language Γ  ontains two unary pr e di ates S, T suh that S ∩ T = ∅ , then Γ stritly implements S ∪ T . Pro of: Let U = S ∪ T . Then U ( x ) + (1 − 1) = S ( x ) + T ( x ) is a strit implemen tation of U ( x ) . ✷ Lemma 4.13 L et H b e a  or e digr aph and Ω an orbit in Aut ( H ) . Then, H stritly implements Ω + and Ω − . Pro of: Assume that H ∈ R D where D = { 1 , 2 , . . . , p } and (without loss of generalit y) assume that 1 ∈ Ω . W e onstrut a strit implemen tation of Ω + ; the other ase an b e pro v ed in a similar w a y . Consider the funtion g ( z 1 , . . . , z p ) = X H ( i,j )=1 H ( z i , z j ) . By om bining the fat that H is a ore with Theorem 1 in [35 ℄, one sees that the follo wing holds: g ( z 1 , . . . , z p ) = | E [ H ] | if and only if the funtion { 1 7→ z 1 , . . . , p 7→ z p } is an automorphism of H . This also implies that a neessary ondition for g ( z 1 , . . . , z p ) = | E [ H ] | is that z 1 is assigned some elemen t in the orbit on taining 1 , i.e. the orbit Ω . W e laim that Ω + an b e stritly implemen ted as follo ws: Ω + ( x ) + ( α − 1 ) = max z ( H ( z 1 , x ) + g ( z )) where z = ( z 1 , z 2 , . . . , z p ) and α = | E [ H ] | + 1 . Assume rst that x ∈ Ω + and  ho ose y ∈ Ω su h that H ( y , x ) = 1 . Then, there exists an automorphism σ su h that σ (1) = y and H ( z 1 , x ) + g ( z ) = 1 + | E [ H ] | b y assigning v ariable z i , 1 ≤ i ≤ p , the v alue σ ( i ) . 20 If x 6∈ Ω + , then there is no y ∈ Ω su h that H ( y , x ) = 1 . If the onstrain t H ( z 1 , x ) is to b e satised, then z 1 m ust b e  hosen su h that z 1 6∈ Ω . W e ha v e already observ ed that su h an assignmen t annot b e extended to an automorphism of H and, onsequen tly , H ( z 1 , x ) + g ( z ) < 1 + | E [ H ] | whenev er z 1 6∈ Ω . Ho w ev er, the assignmen t z i = i , 1 ≤ i ≤ p , mak es H ( z 1 , x ) + g ( z ) = | E [ H ] | sine the iden tit y funtion is an automorphism of H . ✷ Lemma 4.14 If H is a  or e digr aph and Ω an orbit in Aut ( H ) , then, for every k , ther e is a numb er k ′ suh that Max CSP ( { H | Ω } ) - k ≤ AP Max CSP ( { H } ) - k ′ . Pro of: Let V [ H ] = { 1 , 2 , . . . , p } and arbitrarily  ho ose one elemen t d ∈ Ω . Let I = ( V , C ) b e an arbitrary instane of Max CSP ( { H | Ω } ) - k and let V = { v 1 , . . . , v n } . W e onstrut an instane I ′ = ( V ′ ∪ V , C ′ ∪ C ) of Max CSP ( { H } ) - k ′ ( k ′ will b e sp eied b elo w) as follo ws: for ea h v ariable v i ∈ V : 1. A dd fresh v ariables w 1 i , . . . , w p i to V ′ . F or ea h ( a, b ) ∈ E [ H ] , add k opies of the onstrain t H ( w a i , w b i ) to C ′ . 2. Iden tify the v ariables v i and w d i and remo v e v i from V ′ . It is lear that there exist an in teger k ′ , indep enden t of I ′ , su h that I ′ is an instane of Max CSP ( { H } ) - k ′ . Let s ′ b e a solution to I ′ . F or an arbitrary v ariable v i ∈ V , if there is some onstrain t in C ′ whi h is not satised b y s ′ , then w e an get another solution s ′′ b y mo difying s ′ so that ev ery onstrain t in C ′ is satised (if H ( w a i , w b i ) is a onstrain t whi h is not satised b y s ′ then set s ′′ ( w a i ) = a and s ′′ ( w b i ) = b ). W e will denote this p olynomial-time algorithm b y P ′ , so s ′′ = P ′ ( s ′ ) . The orresp onding solution to I will b e denoted b y P ( s ′ ) , so P ( s ′ )( v i ) = P ′ ( s ′ )( w d i ) . The algorithm P ma y mak e some of the onstrain ts in v olving v i unsatised. Ho w ev er, the n um b er of opies, k , of the onstrain ts in C ′ implies that m ( I ′ , s ′ ) ≤ m ( I ′ , P ′ ( s ′ )) . In partiular, this means that an y optimal solution to I ′ an b e used to onstrut another optimal solution whi h satises all onstrain ts in C ′ . Hene, for ea h v i ∈ V , all onstrain ts from step 1 are satised b y s ′′ = P ′ ( s ′ ) . As H is a ore, s ′′ restrited to w 1 i , . . . , w p i (for an y v i ∈ V ) indues an automorphism of H . Denote the automorphism b y f : V [ H ] → V [ H ] and note that f an b e dened as f ( x ) = s ′′ ( w x i ) . F urthermore, s ′′ ( w d i ) ∈ Ω for all w d i ∈ V sine d ∈ Ω . T o simplify the notation w e let l = | E [ H ] | . By a straigh tforw ard probabilisti argumen t w e ha v e opt ( I ) ≥ l p 2 | C | . Using this fat and the argumen t ab o v e w e an b ound the optim um of I ′ as follo ws: opt ( I ′ ) ≤ opt ( I ) + k l | V | ≤ opt ( I ) + 2 k l | C | ≤ opt ( I ) + 2 k p 2 opt ( I ) = (1 + 2 k p 2 ) opt ( I ) . F rom Lemma 2.2 w e kno w that there exists a p olynomial-time appro ximation algorithm A for Max CSP ( H   Ω ) . Let us assume that A is a c -appro ximation algorithm, i.e., it pro dues solutions whi h are c -appro ximate in p olynomial time. W e onstrut the algorithm G in the AP -redution as follo ws: G ( I , s ′ ) =  P ( s ′ ) if m ( I , P ( s ′ )) ≥ m ( I , A ( I )) , A ( I ) otherwise. 21 W e see that opt ( I ) /m ( I , G ( I , s ′ )) ≤ c . Let s ′ b e a r -appro ximate solution to I ′ . As m ( I ′ , s ′ ) ≤ m ( I ′ , P ′ ( s ′ )) , w e get that P ′ ( s ′ ) is a r -appro ximate solution to I ′ , to o. F urther- more, sine P ′ ( s ′ ) satises all onstrain ts in tro dued in step 1, w e ha v e opt ( I ′ ) − m ( I ′ , P ′ ( s ′ )) = opt ( I ) − m ( I , P ( s ′ )) . Let β = 1 + 2 k p 2 and note that opt ( I ) m ( I , G ( I , s ′ )) = m ( I , P ( s ′ )) m ( I , G ( I , s ′ )) + opt ( I ′ ) − m ( I ′ , P ′ ( s ′ )) m ( I , G ( I , s ′ )) ≤ ≤ 1 + opt ( I ′ ) − m ( I ′ , P ′ ( s ′ )) m ( I , G ( I , s ′ )) ≤ ≤ 1 + c · opt ( I ′ ) − m ( I ′ , P ′ ( s ′ )) opt ( I ) ≤ ≤ 1 + cβ · opt ( I ′ ) − m ( I ′ , P ′ ( s ′ )) opt ( I ′ ) ≤ ≤ 1 + cβ · opt ( I ′ ) − m ( I ′ , P ′ ( s ′ )) m ( I ′ , P ′ ( s ′ )) ≤ ≤ 1 + cβ ( r − 1) . ✷ Lemma 4.15 L et H b e a non-empty non-valid digr aph suh that • | V [ H ] | > 2 , • H is a  or e, and • H is not vertex-tr ansitive. Then, either (a) Max CSP ( { H } ) - B is har d to appr oximate, or (b) ther e exists a pr op er subset X of V suh that | X | ≥ 2 , H   X is non-empty, H   X is non-valid and for every k ther e exists a k ′ suh that Max CSP ( { H   X } ) - k ≤ AP Max CSP ( { H } ) - k ′ . Pro of: W e split the pro of in to three ases. Case 1: There exists an orbit Ω 1 ( V [ H ] su h that Ω + 1 on tains at least one orbit. If H   Ω 1 is non-empt y , then w e get the result from Lemma 4.14 sine Ω 1 ( V [ H ] (w e annot ha v e | Ω 1 | = 1 b eause then H w ould on tain a lo op). Assume that H   Ω 1 is empt y . As H   Ω 1 is empt y , w e get that Ω + 1 is a prop er subset of V [ H ] . If H   Ω + 1 is non-empt y , then w e get the result from Lemmas 4.13 , 2.9 and 4.11 . Hene, w e assume that H   Ω + 1 is empt y . Arbitrarily  ho ose an orbit Ω 2 ⊆ Ω + 1 and note that Ω + 1 ∩ Ω − 2 = ∅ sine H   Ω + 1 is empt y . If Ω + 1 ∪ Ω − 2 ( V [ H ] , then w e get the result from Lemmas 4.13 , 2.9 , 4.12 and 4.11 b eause H   Ω + 1 ∪ Ω − 2 is non-empt y . Hene, w e an assume without loss of generalit y that Ω + 1 ∪ Ω − 2 = V [ H ] , and sine Ω + 1 ∩ Ω − 2 = ∅ , w e ha v e an partition of V [ H ] in to the sets Ω + 1 and Ω − 2 . Using the same argumen t as for Ω + 1 , w e an assume that H   Ω − 2 is empt y . Therefore, Ω + 1 , Ω − 2 is a partition of V [ H ] and H   Ω + 1 , H   Ω − 2 are b oth empt y . This implies that H is bipartite and w e get the result from Lemma 4.10 . Case 2: There exists an orbit Ω 1 ⊂ V [ H ] su h that Ω − 1 on tains at least one orbit. This ase is analogous to the previous ase. Case 3: F or ev ery orbit Ω ⊆ V [ H ] , neither Ω + nor Ω − on tains an y orbits. 22 Pi k an y t w o orbits Ω 1 and Ω 2 (not neessarily distint). Assume that there are x ∈ Ω 1 and y ∈ Ω 2 su h that ( x, y ) ∈ E [ H ] . Let z b e an arbitrary v ertex in Ω 2 . Sine Ω 2 is an orbit of H , there is an automorphism ρ ∈ Aut ( H ) su h that ρ ( y ) = z , so ( ρ ( x ) , z ) ∈ E [ H ] . F urthermore, Ω 1 is an orbit of Aut ( H ) so ρ ( x ) ∈ Ω 1 . Sine z w as  hosen arbitrarily , w e onlude that Ω 2 ⊆ Ω + 1 . Ho w ev er, this on tradits our assumption that neither Ω + 1 nor Ω − 1 on tains an y orbit. W e onlude that for an y pair Ω 1 , Ω 2 of orbits and an y x ∈ Ω 1 , y ∈ Ω 2 , w e ha v e ( x, y ) 6∈ E [ G ] . This implies that H is empt y and Case 3 annot o ur. ✷ Lemma 4.16 L et H b e a non-empty non-valid digr aph. Then, Max CSP ( { H } ) - B is har d to appr oximate. Pro of: Due to Lemmas 2.12 and 2.6 , w e an assume that H is a ore. If H is v ertex- transitiv e, then the result follo ws from Lemma 4.9 . If H is not v ertex-transitiv e, then w e an obtain, b y Lemma 4.15 , a smaller graph G su h that G has at least t w o v erties, G is non-empt y , G is non-v alid, and Max CSP ( G ) - B ≤ AP Max CSP ( H ) - B . By rep eatedly using Lemma 4.15 , w e will ev en tually obtain either a graph whi h is v ertex-transitiv e graph or a pro of of appro ximation hardness. In the former ase the result follo ws from Lemma 4.9 . ✷ 4.2 Main Result Armed with the previous lemmas, it is suien t to pro vide an arit y redution argumen t (Lemma 4.17 b elo w) and assem ble the v arious piees to pro v e the main theorem. Lemma 4.17 w as rst pro v ed in [36 ℄ but w e rep eat the pro of here to mak e this rep ort more self-on tained. Lemma 4.17 If R is a non-empty non-valid r elation of arity n ≥ 2 , then R stritly imple- ments a binary non-empty non-valid r elation. Pro of: W e pro v e the lemma b y indution on the arit y of R . The result trivially holds for n = 2 . Assume that the result holds for n = k , k ≥ 2 . W e sho w that it holds for n = k + 1 . Assume rst that there exists ( a 1 , . . . , a k +1 ) ∈ D k +1 su h that R ( a 1 , . . . , a k +1 ) = 1 and |{ a 1 , . . . , a k +1 }| ≤ k . W e assume without loss of generalit y that a k = a k +1 and onsider the prediate R ′ ( x 1 , . . . , x k ) = R ( x 1 , . . . , x k , x k ) . Note that this is a strit 1-implemen tation and R ′ ( d, . . . , d ) = 0 for all d ∈ D . F urthermore, note that R ′ is non-empt y sine R ′ ( a 1 , . . . , a k ) = 1 . Assume no w that |{ a 1 , . . . , a k +1 }| = k + 1 whenev er R ( a 1 , . . . , a k +1 ) = 1 . Consider the prediate R ′ ( x 1 , . . . , x k ) = max y R ( x 1 , . . . , x k , y ) , and note that this is a strit 1-implemen tation. W e see that R ′ ( d, . . . , d ) = 0 for all d ∈ D (due to the ondition ab o v e) and R ′ is non-empt y sine R is non-empt y . ✷ W e are nally able to state the pro of of the main theorem of this setion, Theorem 4.1. Pro of: Let R b e a relation in R ( n ) D . Clearly , Max CSP ( { R } ) an b e solv ed in p olynomial time if R is v alid. If R is empt y , then all solutions ha v e the same measure. Otherwise, if R is non-empt y and not v alid, then w e an, due to Lemma 4.17 , stritly implemen t a binary relation R ′ with R su h that R ′ is neither v alid nor empt y . T ogether with Lemma 4.16 and Lemma 2.9 , w e get the desired result. ✷ W e will no w giv e a simple example on ho w Theorem 4.1 an b e used for studying the appro ximabilit y of onstrain t languages. Consider the follo wing observ ation: Let Γ b e a onstrain t language, R ∈ Γ and Ω an orbit in Aut (Γ) . Then, R   Ω is either d -v alid for ev ery d ∈ Ω or not d -v alid for an y d ∈ Ω . 23 Prop osition 4.18 L et O = { Ω | Ω is an orbit in Aut (Γ) } and let Γ b e a  onstr aint language suh that O ⊆ Γ . If Γ  ontains a k -ary, k > 1 , r elation R that  ontains a tuple ( t 1 , . . . , t k ) suh that R is not t i -valid, for any 1 ≤ i ≤ k , then Max CSP (Γ) is har d to appr oximate. Pro of: W e an view the unary relation U = [ { Ω ∈ O | t i ∈ Ω for some 1 ≤ i ≤ k } as a mem b er of Γ due to Lemma 4.12 . No w, R   U is a non-empt y , non-v alid relation and appro ximabilit y hardness follo ws from Lemmas 2.9 , 4.11 , and Theorem 4.1 . ✷ Corollary 4.19 L et Γ b e a  onstr aint language suh that Aut (Γ)  ontains a single orbit. If Γ  ontains a non-empty k -ary, k > 1 , r elation R whih is not d -valid for al l d ∈ D , then Max CSP (Γ) is har d to appr oximate. Otherwise, Max CSP (Γ) is tr atable. Pro of: If a relation R with the prop erties desrib ed ab o v e exists, then Max CSP (Γ) is hard to appro ximate b y Prop osition 4.18 (note that R annot b e d -v alid for an y d ). Otherwise, ev ery k -ary , k > 1 , relation S ∈ Γ is d -v alid for all d ∈ D . If Γ on tains a unary relation U su h that U ( D , then Aut (Γ) w ould on tain at least t w o orbits whi h on tradit our assumptions. It follo ws that Max CSP (Γ) is trivially solv able. ✷ Note that the onstrain t languages onsidered in Corollary 4.19 ma y b e seen as a general- isation of v ertex-transitiv e graphs. 4.3 Max CSP and Sup ermo dularit y In this setion, w e will pro v e t w o results whose pro ofs mak e use of Theorem 4.1 . The rst result (Prop osition 4.25 ) onerns the hardness of appro ximating Max CSP (Γ) for Γ whi h on tains all at most binary relations whi h are 2-monotone (see Setion 4.3.1 for a denition) on some partially ordered set whi h is not a lattie order. The other result, Theorem 4.27 , states that Max CSP (Γ) is hard to appro ximate if Γ on tains all at most binary sup ermo dular prediates on some lattie and in addition on tains at least one prediate whi h is not sup ermo dular on the lattie. These results strengthens earlier published results [41 , 42 ℄ in v arious w a ys (e.g., they apply to a larger lass of onstrain t languages or they giv e appro ximation hardness instead of NP - hardness). In Setion 4.3.1 w e giv e a few preliminaries whi h are needed in this setion while the new results are on tained in Setion 4.3.2 . 4.3.1 Preliminaries Reall that a partial order ⊑ on a domain D is a latti e or der if, for ev ery x, y ∈ D , there exist a greatest lo w er b ound x ⊓ y and a least upp er b ound x ⊔ y . The algebra L = ( D ; ⊓ , ⊔ ) is a latti e , and x ⊔ y = y ⇐ ⇒ x ⊓ y = x ⇐ ⇒ x ⊑ y . W e will write x ⊏ y if x 6 = y and x ⊑ y . All latties w e onsider will b e nite, and w e will simply refer to these algebras as latti es instead of using the more appropriate term nite latti es . The dir e t pr o dut of L , denoted b y L n , is the lattie with domain D n and op erations ating omp onen t wise. Denition 4.20 (Sup ermo dular funtion) L et L b e a latti e on D . A funtion f : D n → R is  al le d sup ermo dular on L if it satises, f ( a ) + f ( b ) ≤ f ( a ⊓ b ) + f ( a ⊔ b ) (2) for al l a , b ∈ D n . 24 The set of all sup ermo dular prediates on a lattie L will b e denoted b y Spmo d L and a onstrain t language Γ is said to b e sup ermo dular on a lattie L if Γ ⊆ Spmo d L . W e will sometimes use an alternativ e w a y of  haraterising sup ermo dularit y: Theorem 4.21 ([24 ℄) A n n -ary funtion f is sup ermo dular on a latti e L if and only if it satises ine quality (2 ) for al l a = ( a 1 , a 2 , . . . , a n ) , b = ( b 1 , b 2 , . . . , b n ) ∈ L n suh that 1. a i = b i with one ex eption, or 2. a i = b i with two ex eptions, and, for e ah i , the elements a i and b i ar e  omp ar able in L . The follo wing denition rst o urred in [16 ℄. Denition 4.22 (Generalised 2-monotone) Given a p oset P = ( D , ⊑ ) , a pr e di ate f is said to b e generalised 2-monotone on P if f ( x ) = 1 ⇐ ⇒ (( x i 1 ⊑ a i 1 ) ∧ . . . ∧ ( x i s ⊑ a i s )) ∨ (( x j 1 ⊒ b j 1 ) ∧ . . . ∧ ( x j s ⊒ b j s )) wher e x = ( x 1 , x 2 , . . . , x n ) and a i 1 , . . . , a i s , b j 1 , . . . , b j s ∈ D , and either of the two disjunts may b e empty. It is not hard to v erify that generalised 2-monotone prediates on some lattie are su- p ermo dular on the same lattie. F or brevit y , w e will use the w ord 2-monotone instead of generalised 2-monotone. The follo wing theorem follo ws from [ 23 , Remark 4.7℄. The pro of in [23 ℄ uses the orre- sp onding un b ounded o urrene ase as an essen tial stepping stone; see [ 20 ℄ for a pro of of this latter result. Theorem 4.23 ( Max CSP on a Bo olean domain) L et D = { 0 , 1 } and Γ ⊆ R D b e a  or e. If Γ is not sup ermo dular on any latti e on D , then Max CSP (Γ) - B is har d to appr oximate. Otherwise, Max CSP (Γ) is tr atable. 4.3.2 Results The follo wing prop osition is a om bination of results pro v ed in [16 ℄ and [41 ℄. Prop osition 4.24 • If Γ  onsists of 2-monotone r elations on a latti e, then Max CSP (Γ)  an b e solve d in p olynomial time. • L et P = ( D , ⊑ ) b e a p oset whih is not a latti e. If Γ  ontains al l at most binary 2-monotone r elations on P , then Max CSP (Γ) is NP -har d. W e strengthen the seond part of the ab o v e result as follo ws: Prop osition 4.25 L et ⊑ b e a p artial or der, whih is not a latti e or der, on D . If Γ  ontains al l at most binary 2-monotone r elations on ⊑ , then Max CSP (Γ) - B is har d to appr oximate. Pro of: Sine ⊑ is a non-lattie partial order, there exist t w o elemen ts a, b ∈ D su h that either a ⊓ b or a ⊔ b do not exist. W e will giv e a pro of for the rst ase and the other ase an b e handled analogously . Let g ( x, y ) = 1 ⇐ ⇒ ( x ⊑ a ) ∧ ( y ⊑ b ) . The prediate g is 2-monotone on P so g ∈ Γ . W e ha v e t w o ases to onsider: (a) a and b ha v e no ommon lo w er b ound, and (b) a and b ha v e at least t w o maximal ommon lo w er b ounds. In the rst ase g is not v alid. T o see this, note 25 that if there is an elemen t c ∈ D su h that g ( c, c ) = 1 , then c ⊑ a and c ⊑ b , and this means that c is a ommon lo w er b ound for a and b , a on tradition. Hene, g is not v alid, and the prop osition follo ws from Theorem 4.1 . In ase (b) w e will use the domain restrition te hnique from Lemma 4.11 together with Theorem 4.1 . In ase (b), there exist t w o distint elemen ts c, d ∈ D , su h that c, d ⊑ a and c, d ⊑ b . F urthermore, w e an assume that there is no elemen t z ∈ D distint from a, b, c su h that c ⊑ z ⊑ a, b , and, similarly , w e an assume there is no elemen t z ′ ∈ D distint from a, b, d su h that d ⊑ z ′ ⊑ a, b . Let f ( x ) = 1 ⇐ ⇒ ( x ⊒ c ) ∧ ( x ⊒ d ) . This prediate is 2-monotone on P . Note that there is no elemen t z ∈ D su h that f ( z ) = 1 and g ( z , z ) = 1 , but w e ha v e f ( a ) = f ( b ) = g ( a, b ) = 1 . By restriting the domain to D ′ = { x ∈ D | f ( x ) = 1 } with Lemma 4.11 , the result follo ws from Theorem 4.1 . ✷ A diamond is a lattie L on a domain D su h that | D | − 2 elemen ts are pairwise inom- parable. That is, a diamond on | D | elemen ts onsist of a top elemen t, a b ottom elemen t and | D | − 2 elemen ts whi h are pairwise inomparable. The follo wing result w as pro v ed in [42 ℄. Theorem 4.26 L et Γ  ontain al l at most binary 2-monotone pr e di ates on some diamond L . If Γ 6⊆ Spmo d L , then Max CSP (Γ) is NP -har d. By mo difying the original pro of of Theorem 4.26 , w e an strengthen the result in three w a ys: our result applies to arbitrary latties, w e pro v e inappro ximabilit y results instead of NP -hardness, and w e pro v e the result for b ounded o urrene instanes. Theorem 4.27 L et Γ  ontain al l at most binary 2-monotone pr e di ates on an arbitr ary latti e L . If Γ 6⊆ Spmo d L , then Max CSP (Γ) - B is har d to appr oximate. Pro of: Let f ∈ Γ b e a prediate su h that f 6∈ Spmo d L . W e will rst pro v e that f an b e assumed to b e at most binary . By Theorem 4.21 , there is a unary or binary prediate f ′ 6∈ Spmo d L whi h an b e obtained from f b y substituting all but at most t w o v ariables b y onstan ts. W e presen t the initial part of the pro of with the assumption that f ′ is binary and the ase when f ′ is unary an b e dealt with in the same w a y . Denote the onstan ts b y a 3 , a 4 , . . . , a n and assume that f ′ ( x, y ) = f ( x, y , a 3 , a 4 , . . . , a n ) . Let k ≥ 5 b e an in teger and assume that Max CSP (Γ ∪ { f ′ } ) - k is hard to appro ximate. W e will pro v e that Max CSP (Γ) - k is hard to appro ximate b y exhibiting an AP -redution from Max CSP (Γ ∪ { f ′ } ) - k to Max CSP (Γ) - k . Giv en an instane I = ( V , C ) of Max CSP (Γ ∪ { f ′ } ) - k , where C = { C 1 , C 2 , . . . , C q } , w e onstrut an instane I ′ = ( V ′ , C ′ ) of Max CSP (Γ) - k as follo ws: 1. for an y onstrain t ( f ′ , v ) = C j ∈ C , in tro due the onstrain t ( f , v ′ ) in to C , where v ′ = ( v 1 , v 2 , y j 3 , . . . , y j n ) , and add the fresh v ariables y j 3 , y j 4 , . . . , y j n to V ′ . A dd t w o opies of the onstrain ts y j i ⊑ a i and a i ⊑ y j i for ea h i ∈ { 3 , 4 , . . . , n } to C ′ . 2. for other onstrain ts, i.e., ( g , v ) ∈ C where g 6 = f ′ , add ( g , v ) to C ′ . It is lear that I ′ is an instane of Max CSP (Γ) - k . If w e are giv en a solution s ′ to I ′ , w e an onstrut a new solution s ′′ to I ′ b y letting s ′′ ( y j i ) = a i for all i, j and s ′′ ( x ) = s ′ ( x ) , otherwise. Denote this transformation b y P , so s ′′ = P ( s ′ ) . It is not hard to see that m ( I ′ , P ( s ′ )) ≥ m ( I ′ , s ′ ) . F rom Lemma 2.2 w e kno w that there is a onstan t c and p olynomial-time c -appro ximation algorithm A for Max CSP (Γ ∪ { f ′ } ) . W e onstrut the algorithm G in the AP -redution as follo ws: G ( I , s ′ ) =  P ( s ′ )   V if m ( I , P ( s ′ )   V ) ≥ m ( I , A ( I )) , A ( I ) otherwise. 26 W e see that opt ( I ) /m ( I , G ( I , s ′ )) ≤ c . By Lemma 2.2 , there is a onstan t c ′ su h that for an y instane I of Max CSP (Γ) , w e ha v e opt ( I ) ≥ c ′ | C | . F urthermore, due to the onstrution of I ′ and the fat that m ( I ′ , P ( s ′ )) ≥ m ( I ′ , s ′ ) , w e ha v e opt ( I ′ ) ≤ opt ( I ) + 4( n − 2) | C | ≤ opt ( I ) + 4( n − 2 ) c ′ · opt ( I ) ≤ opt ( I ) ·  1 + 4( n − 2 ) c ′  . Let s ′ b e an r -appro ximate solution to I ′ . As m ( I ′ , s ′ ) ≤ m ( I ′ , P ( s ′ )) , w e get that P ( s ′ ) also is an r -appro ximate solution to I ′ . F urthermore, sine P ( s ′ ) satises all onstrain ts in tro dued in step 1, w e ha v e opt ( I ′ ) − m ( I ′ , P ( s ′ )) = opt ( I ) − m ( I , P ( s ′ )   V ) . Let β = 1 + 4 ( n − 2) /c ′ and note that opt ( I ) m ( I , G ( I , s ′ )) = = m ( I , P ( s ′ )   V ) m ( I , G ( I , s ′ )) + opt ( I ′ ) − m ( I ′ , P ( s ′ )) m ( I , G ( I , s ′ )) ≤ ≤ 1 + opt ( I ′ ) − m ( I ′ , P ( s ′ )) m ( I , G ( I , s ′ )) ≤ 1 + c · opt ( I ′ ) − m ( I ′ , P ( s ′ )) opt ( I ) ≤ ≤ 1 + cβ · opt ( I ′ ) − m ( I ′ , P ( s ′ )) opt ( I ′ ) ≤ 1 + cβ · opt ( I ′ ) − m ( I ′ , P ( s ′ )) m ( I ′ , P ( s ′ )) ≤ ≤ 1 + cβ ( r − 1) . W e onlude that Max CSP (Γ) - k is hard to appro ximate if Max CSP (Γ ∪ { f ′ } ) - k is hard to appro ximate. W e will no w pro v e that Max CSP (Γ) - B is hard to appro ximate under the assumption that f is at most binary . W e sa y that the pair ( a , b ) witnesses the non-sup ermo dularity of f if f ( a ) + f ( b ) 6≤ f ( a ⊓ b ) + f ( a ⊔ b ) . Case 1: f is unary . As f is not sup ermo dular on L , there exists elemen ts a, b ∈ L su h that ( a, b ) witnesses the non-sup ermo dularit y of f . Note that a and b annot b e omparable b eause w e w ould ha v e { a ⊔ b, a ⊓ b } = { a , b } , and so f ( a ⊔ b ) + f ( a ⊓ b ) = f ( a ) + f ( b ) on traditing the  hoie of ( a, b ) . W e an no w assume, without loss of generalit y , that f ( a ) = 1 . Let z ∗ = a ⊓ b and z ∗ = a ⊔ b . Note that the t w o prediates u ( x ) = 1 ⇐ ⇒ x ⊑ z ∗ and u ′ ( x ) = 1 ⇐ ⇒ z ∗ ⊑ x are 2-monotone and, hene, on tained in Γ . By using Lemma 4.11 , it is therefore enough to pro v e appro ximation hardness for Max CSP (Γ   D ′ ) - B , where D ′ = { x ∈ D | z ∗ ⊑ x ⊑ z ∗ } . Sub ase 1a: f ( a ) = 1 and f ( b ) = 1 . A t least one of f ( z ∗ ) = 0 and f ( z ∗ ) = 0 m ust hold. Assume that f ( z ∗ ) = 0 , the other ase an b e handled in a similar w a y . Let g ( x, y ) = 1 ⇐ ⇒ [( x ⊑ a ) ∧ ( y ⊑ b )] and note that g is 2-monotone so g ∈ Γ . Let d b e an arbitrary elemen t in D ′ su h that g ( d, d ) = 1 . F rom the denition of g w e kno w that d ⊑ a, b so d ⊑ z ∗ whi h implies that d = z ∗ . F urthermore, w e ha v e g ( a, b ) = 1 , f ( a ) = f ( b ) = 1 , and f ( z ∗ ) = 0 . Let D ′′ = { x ∈ D ′ | f ( x ) = 1 } . By applying Theorem 4.1 to g | D ′′ , w e see that Max CSP (Γ   D ′′ ) - B is hard to appro ximate. No w Lemma 4.11 implies the result for Max CSP (Γ   D ′ ) - B , and hene for Max CSP (Γ) - B . Sub ase 1b: f ( a ) = 1 and f ( b ) = 0 . In this ase, f ( z ∗ ) = 0 and f ( z ∗ ) = 0 holds. 27 T able 1: P ossibilities for g . x y t 1 ( x ) t 2 ( y ) g ( x, y ) 0 0 a 1 b 2 0 0 0 0 1 0 1 a 1 a 2 1 1 0 1 1 1 0 b 1 b 2 1 0 1 1 1 1 1 b 1 a 2 1 0 0 0 0 If there exists d ∈ D ′ su h that b ⊏ d ⊏ z ∗ and f ( d ) = 1 , then w e get f ( a ) = 1 , f ( d ) = 1 , a ⊔ d = z ∗ and f ( z ∗ ) = 0 , so this ase an b e handled b y Sub ase 1a. Assume that su h an elemen t d do es not exist. Let u ( x ) = 1 ⇐ ⇒ b ⊑ x . The prediate u is 2-monotone so u ∈ Γ . Let h ( x ) = f | D ′ ( x ) + u | D ′ ( x ) . By the observ ation ab o v e, this is a strit implemen tation. By Lemmas 2.9 and 2.6 , it is suien t to pro v e the result for Γ ′ = Γ | D ′ ∪ { h } . This an b e done exatly as in the previous sub ase, with D ′′ = { x ∈ D ′ | h ( x ) = 1 } . Case 2: f is binary . W e no w assume that Case 1 do es not apply . By Theorem 4.21 , there exist a 1 , a 2 , b 1 , b 2 su h that f ( a 1 , a 2 ) + f ( b 1 , b 2 ) 6≤ f ( a 1 ⊔ b 1 , a 2 ⊔ b 2 ) + f ( a 1 ⊓ b 1 , a 2 ⊓ b 2 ) (3) where a 1 , b 1 are omparable and a 2 , b 2 are omparable. Note that w e annot ha v e a 1 ⊑ b 1 and a 2 ⊑ b 2 , b eause then the righ t hand side of (3) is equal to f ( b 1 , b 2 ) + f ( a 1 , a 2 ) whi h is a on tradition. Hene, w e an without loss of generalit y assume that a 1 ⊑ b 1 and b 2 ⊑ a 2 . As in Case 1, w e will use Lemma 4.11 to restrit our domain. In this ase, w e will onsider the sub domain D ′ = { x ∈ D | z ∗ ⊑ x ⊑ z ∗ } where z ∗ = a 1 ⊓ b 2 and z ∗ = a 2 ⊔ b 1 . As the t w o prediates u z ∗ ( x ) and u z ∗ ( x ) , dened b y u z ∗ ( x ) = 1 ⇐ ⇒ x ⊑ z ∗ and u z ∗ ( x ) = 1 ⇐ ⇒ z ∗ ⊑ x , are 2-monotone prediates and mem b ers of Γ , Lemma 4.11 tells us that it is suien t to pro v e hardness for Max CSP (Γ ′ ) - B where Γ ′ = Γ   D ′ . W e dene the funtions t i : { 0 , 1 } → { a i , b i } , i = 1 , 2 as follo ws: • t 1 (0) = a 1 and t 1 (1) = b 1 ; • t 2 (0) = b 2 and t 2 (1) = a 2 . Hene, t i (0) is the least elemen t of a i and b i and t i (1) is the greatest elemen t of a i and b i . Our strategy will b e to redue a ertain Bo olean Max CSP problem to Max CSP (Γ ′ ) - B . Dene three Bo olean prediates as follo ws: g ( x, y ) = f ( t 1 ( x ) , t 2 ( y )) , c 0 ( x ) = 1 ⇐ ⇒ x = 0 , and c 1 ( x ) = 1 ⇐ ⇒ x = 1 . One an v erify that Max CSP ( { c 0 , c 1 , g } ) - B is hard to appro ximate for ea h p ossible  hoie of g , b y using Theorem 4.23 ; onsult T able 1 for the dieren t p ossibilities of g . The follo wing 2-monotone prediates (on D ′ ) will b e used in the redution: h i ( x, y ) = 1 ⇐ ⇒ [( x ⊑ z ∗ ) ∧ ( y ⊑ t i (0))] ∨ [( z ∗ ⊑ x ) ∧ ( t i (1) ⊑ y )] , i = 1 , 2 . The prediates h 1 , h 2 are 2-monotone so they b elong to Γ ′ . W e will also use the follo wing prediates: • L d ( x ) = 1 ⇐ ⇒ x ⊑ d , • G d ( x ) = 1 ⇐ ⇒ d ⊑ x , and 28 • N d,d ′ ( x ) = 1 ⇐ ⇒ ( x ⊑ d ) ∨ ( d ′ ⊑ x ) for arbitrary d, d ′ ∈ D ′ . These prediates are 2-monotone. Let w b e an in teger su h that Max CSP ( { g , c 0 , c 1 } ) - w is hard to appro ximate; su h an in teger exists aording to Theorem 4.23 . Let I = ( V , C ) , where V = { x 1 , x 2 , . . . , x n } and C = { C 1 , . . . , C m } , b e an instane of Max CSP ( { g , c 0 , c 1 } ) - w . W e will onstrut an instane I ′ of Max CSP (Γ ′ ) - w ′ , where w ′ = 8 w + 5 , as follo ws: 1. F or ev ery C i ∈ C su h that C i = g ( x j , x k ) , in tro due (a) t w o fresh v ariables y i j and y i k , (b) the onstrain t f ( y i j , y i k ) , () 2 w + 1 opies of the onstrain ts L b 1 ( y i j ) , G a 1 ( y i j ) , N a 1 ,b 1 ( y i j ) , (d) 2 w + 1 opies of the onstrain ts L a 2 ( y i k ) , G b 2 ( y i k ) , N b 2 ,a 2 ( y i k ) , and (e) 2 w + 1 opies of the onstrain ts h 1 ( x j , y i j ) , h 2 ( x k , y i k ) . 2. for ev ery C i ∈ C su h that C i = c 0 ( x j ) , in tro due the onstrain t L z ∗ ( x j ) , and 3. for ev ery C i ∈ C su h that C i = c 1 ( x j ) , in tro due the onstrain t G z ∗ ( x j ) . The in tuition b ehind this onstrution is as follo ws: due to the b ounded o urrene prop- ert y and the quite large n um b er of opies of the onstrain ts in steps 1, 1d and 1e, all of those onstrain ts will b e satised in go o d solutions. The elemen ts 0 and 1 in the Bo olean problem orresp onds to z ∗ and z ∗ , resp etiv ely . This ma y b e seen in the onstrain ts in tro dued in steps 2 and 3. The onstrain ts in tro dued in step 1 essen tially fore the v ariables y i j to b e either a 1 or b 1 , and the onstrain ts in step 1d w ork in a similar w a y . The onstrain ts in step 1e w ork as bijetiv e mappings from the domains { a 1 , b 1 } and { a 2 , b 2 } to { z ∗ , z ∗ } . F or example, h 1 ( x j , y i j ) will set x j to z ∗ if y i j is a 1 , otherwise if y i j is b 1 , then x j will b e set to z ∗ . Finally , the onstrain t in tro dued in step 1b orresp onds to g ( x j , x k ) in the original problem. It is lear that I ′ is an instane of Max CSP (Γ ′ ) - w ′ . Note that due to the b ounded o urrene prop ert y of I ′ , a solution whi h do es not satisfy all onstrain ts in tro dued in steps 1, 1d and 1e an b e used to onstrut a new solution whi h satises those onstrain ts and has a measure whi h is greater than or equal to the measure of the original solution. W e will denote this transformation of solutions b y P . Giv en a solution s ′ to I ′ , w e an onstrut a solution s = G ( s ′ ) to I b y , for ev ery x ∈ V , letting s ( x ) = 0 if P ( s ′ )( x ) = z ∗ and s ( x ) = 1 , otherwise. Let M b e the n um b er of onstrain ts in C of t yp e g . W e ha v e that, for an arbitrary solution s ′ to I ′ , m ( I ′ , P ( s ′ )) = m ( I , G ( s ′ )) + 8 (2 w + 1) · M ≥ m ( I ′ , s ′ ) . F urthermore, opt ( I ′ ) = opt ( I ) + 8(2 w + 1) M . No w, assume that opt ( I ′ ) /m ( I ′ , s ′ ) ≤ ε ′ . It follo ws that opt ( I ′ ) /m ( I ′ , P ( s ′ )) ≤ ε ′ and opt ( I ) + 8(2 w + 1) M m ( I , G ( s ′ )) + 8 (2 w + 1) M ≤ ε ′ ⇒ opt ( I ) ≤ ε ′ m ( I , G ( s ′ )) + ( ε ′ − 1)8(2 w + 1) M ⇒ opt ( I ) m ( I , G ( s ′ )) ≤ ε ′ + 8(2 w + 1) M ( ε ′ − 1) m ( I , G ( s ′ )) . F urthermore, b y standard argumen ts, w e an assume that m ( I , G ( s ′ )) ≥ | C | /c , for some onstan t c . W e get, opt ( I ) m ( I , G ( s ′ )) ≤ ε ′ + 8(2 w + 1) c ( ε ′ − 1) . 29 Hene, a p olynomial time appro ximation algorithm for Max CSP (Γ ′ ) - w ′ with p erformane ratio ε ′ an b e used to obtain ε ′′ -appro ximate solutions, where ε ′′ is giv en b y ε ′ + 8(2 w + 1) c ( ε ′ − 1) , for Max CSP ( { c 0 , c 1 , g } ) - w in p olynomial time. Note that ε ′′ tends to 1 as ε ′ approa hes 1 . This implies that Max CSP (Γ ′ ) - w ′ is hard to appro ximate b eause Max CSP ( { c 0 , c 1 , g } ) - w is hard to appro ximate. ✷ 5 Conlusions and F uture W ork This rep ort ha v e t w o main results: the rst one is that Max CSP (Γ) has a hard gap at lo ation 1 whenev er Γ satises a ertain ondition whi h mak es CSP (Γ) NP -hard. This ondition aptures all onstrain t languages whi h are urren tly kno wn to mak e CSP (Γ) NP - hard. This ondition has also b een onjetured to b e the dividing line b et w een tratable (in P ) CSP s and NP -hard CSP s. The seond result is that single relation Max CSP is hard to appro ximate exept in a few ases where optimal solutions an b e found trivially . It is p ossible to strengthen these results in a n um b er of w a ys. The follo wing p ossibilities applies to b oth of our results. W e ha v e paid no atten tion to the onstan t whi h w e pro v e inappro ximabilit y for. That is, giv en a onstrain t language Γ , what is the smallest onstan t c su h that Max CSP (Γ) is not appro ximable within c − ε for an y ε > 0 in p olynomial time? F or some relations a lot of w ork has b een done in this diretion, f. [ 6℄ for more details. W e ha v e a onstan t n um b er of v ariable o urrenes in our hardness results, but the onstan t is unsp eied. F or some problems, for example Max 2Sa t , it is kno wn that allo wing only three v ariable o urrenes still mak es the problem hard to appro ximate (ev en APX -hard) [ 6℄. This is also true for some other Max CSP problems su h as Max Cut [1 ℄. This leads to the questions: is Max CSP ( { R } ) - 3 hard to appro ximate for all non-v alid non-empt y R ? and, is it true that Max CSP (Γ) - 3 has a hard gap at lo ation 1 whenev er Max CSP (Γ) - B has a hard gap at lo ation 1? One of the main op en problems is to lassify Max CSP (Γ) for all onstrain t languages Γ , with resp et to tratabilit y of nding an optimal solution. The urren t results in this diretion [16 , 23 , 36 , 42 ℄ seems to indiate that the onept of sup ermo dularity is of en tral imp ortane for the omplexit y of Max CSP . Ho w ev er, the problem is op en on b oth ends  w e do not kno w if sup ermo dularit y implies tratabilit y and neither do w e kno w if non- sup ermo dularit y implies non-tratabilit y . Here non-tratabilit y should b e in terpreted as not in PO  under some suitable omplexit y-theoreti assumption, the questions of NP -hardness and appro ximation hardness are, of ourse, also op en. A  kno wledgemen ts. The authors w ould lik e to thank Gusta v Nordh for ommen ts whi h ha v e impro v ed the pre- sen tation of this pap er. 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