Solving Linear Constraints in Elementary Abelian p-Groups of Symmetries

Solving Linear Constrain ts in Elemen tary Ab elian p -Groups of Sy mmetries Thierry Bo y de la T our & Mnac ho Ec henim Lab oratoire d’Informatique de Grenoble, CNRS - Grenoble I NP Bˆ atimen t IMAG C - 220 rue de la Chimie F-38400 S ain t-Martin-d’H` eres email: thierry.boy-de-la -tour@imag.f r , mnacho.ech enim@imag.f r Abstract Symmetries occur naturally in CSP or SA T problems and are not very difficult to disco ver, but u sing them to prune th e searc h space tends to b e very chal lenging. Indeed, this usually requires finding sp ecific elements in a group of symmetries that can be huge , and the problem of their very existence is NP-hard. W e form ulate such an existence problem as a constrain t problem on one v ariable (the symmetry to be used) ranging o ver a group, and try to fi nd restrictions that may be solv ed in polynomial time. By considering a simple form of constraints (restricted by a cardinality k ) and th e class of groups th at hav e the structure of F p -vector spaces, w e prop ose a partial algorithm based on linear algebra. This p olynomial algorithm alw a ys applies when k = p = 2, b u t ma y fail otherwise as w e prov e the problem to b e NP-hard for all other va lues of k and p . Exp erimen ts sho w that this approach though restricted should allo w for an efficien t use of at least some groups of symmetries. W e conclude with a few directions to b e ex plored t o efficiently solve th is problem on the general case. keyw ords: sy mmetries, linear a lg ebra, complexity 1 In tro duction Symmetries are permutations o f input sy m bols that, when applied to an instance of a computationa l problem, leave its solution inv ariant. Since na ¨ ıve algorithms repro duce these symmetr ie s in their s e arch space, it is tempting to use them as a pruning device. A typical example is the pigeon-hole problem in prop ositional logic: it s symmetries show that an y pigeon can be swapp ed with any other (and similarly for the holes ), and therefor e play equiv alent rˆ oles. Since in v ariance is s ta ble by co mposition, the set of symmetries of an instance forms a p ermutation g roup. This means that the information they provide has a ma thema tical str uc tur e that should yield nice computing pr operties (even if 1 they o ccur only at the meta-level). In fact, some techniques from computational group theory hav e been employ ed to discov er symmetries and to use them. How ev er, it seems that algor ithms c a n use symmetries in a straightforward wa y only if their search space preserves the s tructure of the group of symmetries in some sense (subtrees should somehow c o rresp ond to subgroups ). This is usually not the c a se for the most efficient algo rithms developed in the field of A I. Hence man y different methods ha ve b een de velop ed in order to prune the search space with symmetries, either by desig ning sp ecial a lgorithms or by mo difying instance s throug h symmetry-br eaking. Much work has b een devoted to this sub ject, see e .g. [12] and the reference s ther ein. One feature common to these metho ds is that they ideally assume the ability to pro duce symmetries that have particular pr operties , suitable fo r the pruning scheme. But this problem also happens to be NP -hard, which e x plains wh y symmetries that do not re s ult in the b est pruning may b e used. The time sp en t on searching and using symmetr ie s do es not alwa ys pay off. This sug gests that the gro up structure may no t be sufficient to induce enough computational prop erties to ens ur e efficient pr uning. Our aim in this pap er is to in vestigate ways of finding suitable symmetries in p olynomial time. T o this purp ose we first formulate this se arch problem by a la nguage of constr a in ts as simple as p ossible. This is the to pic of Sectio n 3. W e then c o nsider res tr ictions of this sea rc h problem that co nfer deep er math- ematical structure to the groups. The idea is to transform the constraints into line ar e quations , whic h would then b e easy to solve b y means of basic computer algebra. T o the b est o f o ur knowledge, a lthough restrictio ns to vector spaces hav e alr eady been cons ider ed in the literature, such a transformation repr esen ts a novel approach. This means that we need to as sume that the groups a re also vector spaces, and to develop wa ys of efficiently working with s ymmetries as vectors (i.e., essentially of co mputing their co ordinates in a suitable basis). This is developed in Sections 4 and 5, leading to a p olynomial a lgorithm that solves so-called linear constra ints. W e w ill then see in Section 6 that, even with the simple co nstraint s and vector spaces, the search problem r emains NP-hard in mos t cases. Ex p eriments in Section 7 illustrate the efficiency of this p olyno- mial algo rithm on ra ndom sa mples of linea r c o nstraint s, compared to a gene r al purp ose algorithm. W e suggest in the c o nclusion a few dir ections for using this approach in a wider setting. 2 Definitions W e do no t r ecall the most basic definitions a nd notations fro m group theory or linear algebra, suc h as cyc le s or bases, which can be found in standard textb o oks, e.g. [3], exc e pt in order to se ttle notations . Given a finite set A , we denote by Sym( A ) the group of p erm utations of A . If g 1 , . . . , g m are p erm utations of A , we de no te b y ⌈ g 1 , . . . , g m ⌉ the subgroup of Sym( A ) genera ted by these per m utations. F or a ∈ A and g , g ′ ∈ Sym( A ), the image of a by g is deno ted by a g , and the comp osition of p erm utations g ′ ◦ g 2 by g g ′ , so that a gg ′ = ( a g ) g ′ . F r om a computational p oin t of view, it is o b vious that the pro duct g g ′ can b e p erformed in time linear in | A | . The or der of g is the smalle st pos itive integer n such that g n is the ident ity . Let G be a p erm utation g roup on A , the orbit of a in G (or G -or bit of a ) is a G = { a g | g ∈ G } . The set of G -orbits forms a partition of A , denoted by OP ( G ). The g r oup G is tr ansitive if it has only one o r bit (i.e., O P ( G ) = { A } ). It is easy to see that OP ( ⌈ g ⌉ ) can be o bta ined fro m the cycles of g , e.g . if A = { 1 , . . . , 6 } then O P ( ⌈ (1 2)(3 4 5) ⌉ ) = {{ 1 , 2 } , { 3 , 4 , 5 } , { 6 }} . The refinement o rder on partitions of A ( P ⊑ P ′ iff ∀ O ∈ P , ∃ O ′ ∈ P ′ s.t. O ⊆ O ′ ) is a complete lattice; the least upper b ound P ⊔ P ′ is obtained by merging the non-disjoint elements of P and P ′ . The smallest partition is ⊥ A = {{ a } | a ∈ A } a nd the g reatest is ⊤ A = { A } . Given tw o p erm utation g r oups G a nd G ′ on A , O P ( ⌈ G ∪ G ′ ⌉ ) = O P ( G ) ⊔ O P ( G ′ ) (s e e [1, chap. 7]). Hence, starting with m gener ators g 1 , . . . , g m the orbit partition O P ( ⌈ g 1 , . . . , g m ⌉ ) = F m i =1 OP ( ⌈ g i ⌉ ) ca n b e computed in time p olynomial in m and | A | . A group G is an elementary Ab elian p -g roup if it is Abelian and its non-triv ial elements hav e order p , a prime num ber. I t is simple to test this property on the gener ators of a g roup: G is a n elementary Abelian p -gr o up iff its gene r ators commute a nd hav e o rder p . If this is the case we adopt the a dditiv e notation, e.g. ( a b ) + ( c d ) = ( a b )( c d ), 2 ( a b c ) = ( a b c ) 2 = ( a c b ) and 0 is the ident ity . By considering m ultiplication by a n in teger as a n e xternal product on the set of integers mo dulo p , we confer to G the structur e of an F p -vector space. Co n versely , every F p -vector space is isomorphic to an element ary Ab elian per m utation p -gr oup. Since the class of elementary Abelia n p -groups is closed under homomorphic images and for all O ∈ OP ( G ) the restriction to O op erator is a morphism from G to Sy m( O ), if G is a n F p -vector space then so is its image G | O = { g | O | g ∈ G } (although it may not b e a subgro up of G ). The gr oups G | O are the tr ansitive c onst itu ents o f G . In the sequel, unless stated otherwise, a , b, c denote member s of A , u, v , w denote vectors (i.e., p ermut ations o n A ), h and f denote bases of vector spaces, and the x i ’s co or dina tes in vector spaces (i.e., integers modulo p ). 3 Constrain ts on symmetries As mentioned a bov e, techniques designed to compute gener ators for a g roup of symmetries ar e well-known. Basic a lly , they consist in building a graph with the elements to b e p erm uted as vertices, and with edges and la bels that dep end on the instance, so that its automorphism gro up is the exp ected gro up of symme- tries. Building this graph is usually straightforward a nd its automo rphism group can then b e computed with e.g. the well-known prog ram nauty [8]. Thoug h not po lynomial, this algorithm has a low av erage complexity and is very efficient. W e thus assume given a group of symmetries that is spec ified b y a g enerating set of m p ermu tations g 1 , . . . , g m of a set A . The gro up G = ⌈ g 1 , . . . , g m ⌉ is the set in which a p ermutation g s atisfying a giv en constr a in t is sear c hed for. In 3 this section we define a language for expressing constraints on g that is b oth useful and simple. W e b egin with an exa mple. W e consider the problem mmc f rom [2]. Given a model M of a formula built on pr opos itio nal v ar iables which a re linea rly order ed (say , x < y < z ), g M b e ing the interpretation defined by g M ( a ) = M ( a g ), a nd given a gr oup G by its gene r ators, the problem mmc consis ts in deciding whether there exists a g ∈ G such that g M < M (interpretations are ordered lexicogra phically). This problem is requir ed for computing symmetr y -breaking predica tes, and is s ho wn in [2] to b e NP- complete. W e can translate g M < M a s: M ( x g ) M ( y g ) M ( z g ) < M ( x ) M ( y ) M ( z ), and then as M ( x g ) < M ( x ) ∨ M ( x g ) = M ( x ) ∧ [ M ( y g ) < M ( y ) ∨ M ( y g ) = M ( y ) ∧ M ( z g ) < M ( z )] . A tra ns lation into disjunctive normal form yields: M ( x g ) < M ( x ) ∨ M ( x g ) = M ( x ) ∧ M ( y g ) < M ( y ) ∨ M ( x g ) = M ( x ) ∧ M ( y g ) = M ( y ) ∧ M ( z g ) < M ( z ) . If X denotes { a | M ( a ) < M ( x ) } then the first disjunct transla tes in to x g ∈ X . Similarly , let X ′ = { a | M ( a ) = M ( x ) } and Y = { a | M ( a ) < M ( y ) } , the second disjunct is x g ∈ X ′ ∧ y g ∈ Y , etc. Thus the problem mmc can be expressed by means of bo olean combinations of atomic c onstr aints of the for m x σ ∈ X , where σ is the (unique) v ar iable rang ing in G . Note that the negation o f any atomic constraint can b e expressed as an atomic constraint 1 (with the complement set), hence negation c an be r uled out from the la ng uage. Since the set of solutions of a disjunction is the union of the s olutions of e a c h disjunct, we fo cus on s olving conjunctions of atomic c onstraints. Definition 1 We addr ess the c omputational pr oblem gc , for Gr oup Constr aint, that takes as input a set A of c ar dinality n , p ermutations g 1 , . . . , g m of A , and a c onstr aint ϕ which is a c onjunction of atomic formulas of t he form x σ ∈ X , wher e σ is a u nique variable and x ∈ A, X ⊆ A . The decision problem gc c onsist in che cking for the existenc e of a p ermutation g ∈ ⌈ g 1 , . . . , g m ⌉ t hat satisfies al l the c onjuncts in ϕ (a c onjunct x σ ∈ X i s satisfied by g if x g ∈ X ). The asso ciate d s earch problem is the c omput ation of such a g if it exists. A num b er of transforma tions o n the cons train t that preser ve the s e t of so- lutions can b e applied. These are: x σ ∈ X ∧ x σ ∈ Y − → x σ ∈ X ∩ Y , ϕ − → ϕ ∧ x σ ∈ x G . The correctness of the first transformation is obvious and that of the sec ond trivial since x g ∈ x G holds for all g ∈ G . By a pply ing these tr a nsformations 1 This is no longer true if we break down x g ∈ { x 1 , . . . , x n } into x g = x 1 ∨ . . . ∨ x g = x n , hence atomic constrain ts of the form x g = y would not necessaril y be simpler to handle. 4 systematically (tog ether with associa tivit y-commutativit y o f co njunction), as - suming tha t A = { a 1 , . . . , a n } it is alwa ys p ossible to transfor m any given con- straint in to an equiv alent constraint of the form a σ 1 ∈ A 1 ∧ . . . ∧ a σ n ∈ A n , where A i ⊆ a G i for all 1 ≤ i ≤ n . Computing this norma l form is linea r in the length of ϕ a nd | A | . This normal form may be represented as a function C fro m A to 2 A , where C ( a i ) = A i . The problem gc then c o nsists in finding a g ∈ G such that ∀ a ∈ A, a g ∈ C ( a ), and we may cons ider C dir e ctly as a n input of the pro blem, so that an ins ta nce of gc is a tuple h A, g 1 , . . . , g m , C i (or h A, G, C i in short). W e say that C is a k -c onst r aint if ∀ a ∈ A, |C ( a ) | ≤ k . As mentioned ab ov e, in the sequel we o nly co nsider, for every pr ime p , the restriction of gc to the instances where G = ⌈ g 1 , . . . , g m ⌉ is an elementary Abelia n p -gro up, i.e., an F p -vector s pace 2 . 4 F rom p erm utations to linear algebra Before giving the technical details of the transfor mation from a problem on per m utations to a problem in linear a lgebra, w e briefly summarize the way we pro ceed. Since our a pproach is based on the transitive constituents G | O of G (for O ∈ O P ( G )) , we first define a vector s pa ce F that contains as subs pa ces bo th the group G and the G | O ’s. W e then prov e a fundamental pro perty of the G | O ’s and use it to realize a p olynomial test of linear dep endence r estricted to these subspaces. This is used to compute a bas is of each G | O and to obtain the co ordinates of any u ∈ G | O in this basis , in po lynomial time. This directly yields a basis of F and the c oor dinates o f any u ∈ F in this bas is . It is then standa r d to tr ansform any linear v ariety of F , such as G , as a set of linear equations using basic linear algebra . Sectio n 5 will b e devoted to a pplying thes e techniques to the cons train t C . In the sequel we write P for the o rbit partition O P ( G ). 4.1 The sup er-space F The gro ups G and the G | O ’s are of cour se all included in Sym( A ), but Sym( A ) is not elementary Abelia n hence not a vector space. Let F b e the p erm utation group on A generated by the transitive constituen ts of G , i.e., F = ⌈ S O ∈P G | O ⌉ . Lemma 1 F is an F p -ve ctor sp ac e that c ontains G and F = M O ∈P G | O . Pr o of. F is generated by the union of generating sets of the g roups G | O , i.e., by the set { g i | O | 1 ≤ i ≤ m, O ∈ P } . Since or bits ar e mutually disjoint and 2 Note tha t solving suc h constrain ts is m uc h simpler than computing an y lex-leader formu la as in [7], where restrictions to vector spaces are already considered. O ur const raints are therefore not relev an t to the pr oblem of building lex-leader formulas, and are not m ean t to be used in the symmetry-breaking sc heme of [2]. 5 per m utations on disjoint sets commute, g i | O g j | O ′ = g j | O ′ g i | O if O 6 = O ′ . F ur- thermore g i | O g j | O = ( g i g j ) | O = ( g j g i ) | O = g j | O g i | O since G | O is Ab elian. Hence F is Ab elian, and similarly its non- tr ivial e le men ts hav e order p . W e therefor e use the additive notation in F , which y ie lds F = P O ∈P G | O . F or any O ∈ P , let A ′ = A \ O and F O = P O ′ 6 = O G | O ′ , since F O is a subg roup of Sym( A ′ ), then G | O ∩ F O ⊆ Sym( O ) ∩ Sym( A ′ ) = { 0 } . This pr o ves that F is the (internal) direct sum of the G | O ’s, which is written F = L O ∈P G | O . It is clear that G is a subspace of F since an y u ∈ G can b e written as u = P o ∈P u | o and hence b elongs to F . ✸ W e ca ll F the sup er-sp ac e o f G . Since the sum in Lemma 1 is direct, the decomp osition of a n y vector u in F as a sum o f element s of the G | O ’s is unique, the dimension d o f F is the sum of the dimensio ns d O of G | O for O ∈ P , and a basis for F c an be obtained by concatenating bases for the subspace s G | O . 4.2 Orbits as affine spaces F or all O ∈ P , the tr ansitiv e constituent G | O is o f cour se transitive in O . This trivial fact yields a fundamental prop erty: Lemma 2 ( O , G | O ) is an affine s p ac e. Pr o of. W e define the external sum + : O × G | O → O , for all a ∈ O and u ∈ G | O , by a + u = a u (the ima g e of a by u ) and prov e that the tw o ax ioms o f affine spaces hold. F or all v ∈ G | O it is clear that ( a + u ) + v = ( a u ) v = a ( u + v ) = a + ( u + v ) . Consider φ a : G | O → O such that φ a ( u ) = a + u , we pr o ve that φ a is bijective. It is obviously onto since G | O is transitive on O : ∀ b ∈ O , ∃ u ∈ G | O such that b = a u = φ a ( u ). Ass ume now that φ a ( u ) = φ a ( v ), i.e., a u = a v , then for a n y b ∈ O , if w ∈ G | O is such that a w = b , then b u = a ( w + u ) = a ( u + w ) = ( a u ) w = ( a v ) w = b v , hence u = v , and φ a is injective. ✸ This is nothing mor e than a geo metric interpretation of a kno wn result: that transitive Ab elian groups are “r egular” (see [10, theorem 10.3.4 ]). This entails that | O | = | G | O | , and that the external sum is r e gular , i.e., an equality a + u = b determines every term from the tw o other s . In particular the unique vector u ∈ G | O such that a + u = b is us ually written u = b − a (or − → ab ), and we also write b = a − u for b = a + ( − u ), which is equiv a len t to b + u = a . Note that there is one ex ternal sum (a nd difference) per orbit, but s ince they ar e disjoint it is unambiguous to denote them with the same symbol. In the sequel w e use the additiv e no tation on sets S, S ′ of vectors or elements of a n o rbit, i.e., if ε ∈ { + , −} then S εS ′ = { sεs ′ | s ∈ S, s ′ ∈ S ′ } . If one set is a singleton, say S = { s } , we write sεS ′ for { s } εS ′ . This is o f cours e compatible with the notations already used when S and S ′ are subspaces of F . 6 4.3 Computing a basis for F Since G | O is a finite F p -vector space, its car dinalit y m ust be p d O . But this cardinality is also that of O which is kno wn, hence the dimension d O can b e computed: d O = log p | O | . F urthermore, since g 1 | O , . . . , g m | O is a gener ating set for G | O , it is p ossible to extract a ba s is f O of G | O from this set by disc a rding m − d O linearly dep enden t vectors. F or this a test of linear dep endence is required. Lemma 3 F or any subsp ac e H of G | O , any u ∈ G | O and any a ∈ O , u ∈ H iff a u ∈ a H . Pr o of. By regularity o f the external sum, u ∈ H iff a + u ∈ a + H . By definition a + u = a u and a + H = { a + v | v ∈ H } = { a v | v ∈ H } = a H . ✸ Linear dependence is ther efore re duce d to computing the orbit of an arbitrary po in t a ∈ O . A bas is of G | O can b e built step b y step b y co mputing the corres p onding orbit partition Q of O , with the following function B:    B([ ] , h , Q ) = h , B([ u ]@ l , h , Q ) = B( l, h , Q ) if a u ∈ a [ Q ] , = B( l , [ u ]@ h , O P ( u ) ⊔ Q ) otherwise. Here l is a list of vectors, @ is the conca tenation of lists and [ ] is the empt y list. Lemma 4 B([ g 1 | O , . . . , g m | O ] , [ ] , ⊥ O ) is a b asis of G | O . Pr o of. Let f O = B([ g 1 | O , . . . , g m | O ] , [ ] , ⊥ O ). Since ⊥ O is the orbit partition of the trivia l gr oup { 0 } on O , the inv ariant Q = OP ( ⌈ h ⌉ ) is maintained throughout the c o mputation. This mea ns that the class a [ Q ] of a mo dulo Q is the orbit a ⌈ h ⌉ , and that a u ∈ a [ Q ] is equiv alen t to u ∈ ⌈ h ⌉ by Lemma 3. Hence u is added to h if and o nly if it is linea rly indep enden t from the latter, which prov es that h remains free. This also pr o ves that ⌈ l @ h ⌉ is inv ariant, hence ⌈ f O ⌉ = ⌈ g 1 | O , . . . , g m | O ⌉ = G | O and f O is fr e e , it is thus a basis of G | O . ✸ Example. Let O = { 1 , . . . , 8 } and g 1 = (1 2 )(3 4)(5 6 )(7 8) , g 2 = (1 5 )(2 6)(3 7 )(4 8) , g 3 = (1 3 )(2 4)(5 7 )(6 8) , W e choose a = 1, then f O = B([ g 1 , g 2 , g 3 ] , [ ] , ⊥ O ) = B([ g 2 , g 3 ] , [ g 1 ] , O P ( g 1 )) since 1 g 1 = 2 6∈ 1 [ ⊥ O ] = { 1 } = B([ g 3 ] , [ g 2 , g 1 ] , O P ( g 2 ) ⊔ O P ( g 1 )) since 1 g 2 = 5 6∈ 1 [ OP ( g 1 )] = { 1 , 2 } = B([ ] , [ g 3 , g 2 , g 1 ] , ⊤ O ) since 1 g 3 = 3 6∈ 1[ O P ( g 2 ) ⊔ O P ( g 1 )] = { 1 , 2 , 5 , 6 } = [ g 3 , g 2 , g 1 ] . 7 ✸ Of co urse the algo rithm can be interrupted once h has d O elements (or equiv alently when Q = ⊤ O ). Building f O requires at most m rec ur siv e calls a nd the co mputatio n of exactly d O ≤ m or bit partitions of O , each being p olynomial in | O | ≤ n , hence f O is computed in time p olynomial in n and m . The bases f O can be concatenated (in an arbitra ry order) to form a basis f of F . The length d of this basis may b e greater than m , but since d O = log p | O | ≤ | O | p , necessarily d = X O ∈P d O ≤ X O ∈P | O | p = n p , hence computing f is ag ain po lynomial in n and m . 4.4 Computing co ordinates in the basis for F The co ordina tes of any vector u ∈ F in basis f can b e obtained by computing, for all O ∈ P , the co ordinates of u | O ∈ G | O in the basis f O , a nd by concatenating these co ordina tes in the sa me order as the one used to build f . W e show how to compute the co ordina tes in f O of the p ermu tations in G | O . Since f O is a basis of G | O , ther e is a 1-1 corresp ondence fr o m the tuples h x 1 , . . . , x d O i ∈ ( F p ) d O to the elements of G | O , given by P d O i =1 x i h i , which can be computed in polyno mial time: each x i h i requires comp osing x i − 1 < p times the p erm utation h i of O with itself, hence computing a p erm utation from its co ordinates in f O can be computed in time O( d O p | O | ), which is bo unded by O( n log n ) (s ince p is a constant). Computing the co ordinates in f O of a given u ∈ G | O means c o mputing the inv e r se of the previous corr espondence. T his can b e p erformed by browsing through a ll po ssible v alues h x 1 , . . . , x d O i ∈ ( F p ) d O un til P d O i =1 x i h i equals u . Since | O | = | G | O | = p d O , this r e quires at most | O | iter ations, hence can b e computed in time O( d O p | O | 2 ), which is b ounded by O( n 2 log n ). The same technique may be applied if u is given as b − a , where a, b ∈ O . In this ca se it is necessary to chec k whether b = a + P d O i =1 x i h i , i.e., whether b is the image of a by the p ermutation P d O i =1 x i h i . The complexity is th us the same as ab o ve. Note that this also a llo ws to compute b − a explicitly a s a pe r m utation. F rom a practical p oint of view we s hould av oid rep eated computations of per m utations f rom co ordinates: this could be done by stor ing v alues in a suitable array . Using our geo metric interpretation, we cho ose arbitrarily an origin a ∈ O and define the co ordinates of any point b ∈ O as tho s e of the vector b − a relative to f O . W e can therefor e fill an ar ray that ass ociates its co ordinates to each entry b ∈ O , by browsing through all p ossible coo rdinates a s explaine d ab o ve. Since this a rray ha s | O | entries, filling it takes p o lynomial time. Then, given a p erm utation u , we need only compute the ima ge b = u a and pick the co ordinates of b in the ar ray; these ar e the co ordinates of u . 8 In the sequel we write f = f 1 , . . . , f d , and for any u ∈ F , if u = P d i =1 x i f i where the x i ∈ F p are the co ordinates of u in f , we write u f = t ( x 1 · · · x d ) the column matrix of these co ordinates . 4.5 A c haracterization of linear v arieties in F Since the sup er-space F is isomorphic to the vector space ( F p ) d , and this iso - morphism can b e co mputed (through the co ordinates in f ) in b oth directions, computations with matrice s can b e substituted for computations with p ermu- tations. This means that standard algor ithms from linea r algebr a apply , in particular Gaussian elimination. Note that exact computations ca n b e p er- formed in F p , including division, in time at most quadratic in the num b er of bits (see, e.g. [5, p.1 17]), hence in constant time in the pr e s en t context. In particular, it is now s traight forward to test whether a family of l ≤ d vectors u 1 , . . . , u l ∈ F is line a rly dep enden t, by first computing the co ordinates u i = P d j =1 x ij f j and then by perfor ming Gaussia n elimination on the l × d - matrix ( x ij ) (which requires a num ber of op erations at most cubic in d ). The family is linearly dependent iff Gaussian elimination yields a zero ro w in the resulting matrix (the num ber of no n-zero lines after Gaussian elimination is the rank of the matr ix, i.e., the dimension o f the spa ce ⌈ u 1 , . . . , u l ⌉ ). Assume we are given a line ar variety v + H of F by the p ermutation v and a genera ting set for the subspace H . W e ca n compute the co ordinates in f of these permutations and, using the previous pro cedure, extract a basis h 1 , . . . , h d ′ of H from the gener ators of H , whe r e d ′ is the dimension of H . The vectors h 1 , . . . , h d ′ together with the v ectors in f form a generating family of F ; the free family h 1 , . . . , h d ′ can therefore b e completed into a ba sis h = h 1 , . . . , h d of F by adding d − d ′ vectors taken from f . If P deno tes the ma trix whose i th column is ( h i ) f then P is the change of basis matrix from h to f : u f = P u h for all u ∈ F . This matrix is inv ertible and its inv erse can b e co mputed using the Gauss-Jo r dan algo rithm in time cubic in d . This means that the co ordinates of any vector u in h (or in any ba s is) ca n be co mputed in p o lynomial time through u h = P − 1 u f . Mem ber s hip of u to the subspace H may b e c hec ked simply by making sure the la st d − d ′ co ordinates of u h are equal to zero. This can b e expres s ed by building the diagonal d × d -matrix D with ones on the last d − d ′ po sitions of the diag o nal a nd z eroes elsewhere: D =  0 0 0 I  , where I is the ( d − d ′ ) × ( d − d ′ ) ide ntit y matrix. Th us vector u ∈ F b elongs to H iff D u h = 0. Let M H = D P − 1 , we hav e shown that: Lemma 5 F r om any line ar variety v + H of F c an b e c ompute d in p olynomia l time a d × d -matrix M H such that ∀ u ∈ F , u ∈ v + H i ff M H u f = M H v f . 9 Conv ersely , it is well kno wn that the set of solutions u of a ny system of linear equations on d unknowns (the co ordinates o f u in f ) is either empty or a linear v ar ie ty of F . 5 Solving linear constrain ts The vector space G b eing a linear v ar iet y of F b y Lemma 1, its elements are characterized a s the s olutions u o f a sy stem of linea r eq ua tions M G u f = 0, a s shown in Lemma 5 (by taking v = 0). W e now inv estigate how to character iz e the cons train t C by another s y stem of linear equations. 5.1 Constrain ts as sets of vectors The firs t step tow ards this characterization is to ex plicitly represent the set o f vectors u ∈ E that satisfy a constraint C . Let V O = \ a ∈ O C ( a ) − a for all O ∈ P . Lemma 6 A ve ctor u ∈ F satisfies C iff u ∈ X O ∈P V O . Pr o of. Assume a vector u ∈ F satisfies the constraint C , i.e., for all a ∈ A , a u ∈ C ( a ). Then for all O in the o rbit partition P a nd for all a ∈ O , since a u = a u | O ∈ O and ( O , G | O ) is an affine space, we may wr ite a + u | O ∈ C ( a ), and since C ( a ) ⊆ O , this is equiv alent to u | O ∈ C ( a ) − a . But this is true for all a ∈ O , hence u | O ∈ V O . Since u = P O ∈P u | O , vector u must belong to P O ∈P V O . Conv ersely , let u ∈ P O ∈P V O and let a ∈ A . If O = a G then u | O ∈ V O , hence in particular u | O ∈ C ( a ) − a , i.e., a u = a + u | O ∈ C ( a ). Hence u s atisfies the cons train t C . ✸ The sets V O can b e computed fo r each O ∈ P by selecting an arbitrar y a ∈ O and computing the co ordina tes of c − a in f O for a ll c ∈ C ( a ), and this op eration is p olynomial in n for each c as mentioned in se c tio n 4.4. Pr ovided C is a k -constraint this yields at most k elements in C ( a ) − a . Then, V O is the set of those vectors u from C ( a ) − a that also belo ng to C ( b ) − b for all b ∈ O \ { a } , i.e., such that b u ∈ C ( b ), which requires tha t u be also computed a s an explicit per m utation. Computing the co ordina tes of all the elements of V O is therefore po lynomial in n and k . 5.2 Linear constrain ts The size of the set P O ∈P V O is Q O ∈P | V O | . If one o f the sets V O is empty , then obviously co nstraint C is unsatisfiable. But in genera l this s et is exp onen tial in the num be r o f G -or bits (and therefore in n , the worst ca se b eing 2 n 2 with n 2 orbits o f size 2). This motiv a tes the following definition. 10 Definition 2 C is a linear co nstrain t if X O ∈P V O is either empty or a line ar variety of F . In order to c heck whether a constraint C is linear or not, w e provide a simple characterization of this prop erty . F or O ∈ P , let w O be a n ar bitrary e le men t of V O and E O = V O − w O . Lemma 7 X O ∈P V O is a line ar variety of F iff ∀ O ∈ P , dim ⌈ E O ⌉ = log p | V O | . Pr o of. Since F = L O ∈P G | O and V O ⊆ G | O , it is clear tha t P O ∈P V O is a linear v ariety of F iff V O is a linea r v ariety of G | O for all O ∈ P . If this is so then E O is a subspace o f G | O that do es not dep end on w O . Hence this is equiv alent to all the E O ’s b eing subspaces, hence to ⌈ E O ⌉ = E O . Ag a in this is equiv alent to p dim ⌈ E O ⌉ = |⌈ E O ⌉| = | E O | = | V O | . ✸ The dimension of ⌈ E O ⌉ is the rank o f the matrix for med by the coo rdinates of the vectors in E O , whic h ca n be computed in p olynomial time and co mpared to log p | V O | . Computing this r ank is of course useless if log p | V O | is no t an int eger. Hence the linea rit y of C can b e tested in time p olynomial in n (since |P | ≤ n ). Note that the case where V O is a single ton corr esponds to E O being reduced to the trivia l subspace { 0 } of dimension 0. Let w = P O ∈P w O and E = L O ∈P E O for every O ∈ P , so that P O ∈P V O is the linear v ariety w + E (as suming it is not empty). Theorem 8 If C is line ar then the pr oblem gc is e quivalent to a syst em of line ar e quations on c o or dinates of a solution in f , and this system c an b e c ompute d and solve d in p olynomia l time. Pr o of. If P O ∈P V O = ∅ then the instance h A, G, C i of gc has no so lution, which is equiv a len t to the linear equation 0 = 1. Other wise P O ∈P V O = w + E and by Lemma 6 any u ∈ F is a solution of the instance h A, G, C i iff u ∈ G and u ∈ w + H , whic h is equiv a len t by Lemma 5 to  M G u f = 0 M E u f = M E w f . This is a sy stem of 2 d linear e q uations on d unknowns, it can be solved b y Gaussian elimination in time cubic in d . ✸ Corollary 9 The pr oblem gc r estricte d to p = k = 2 is p olynomial. Pr o of. Ev ery non-empty V O has at mos t k = 2 elements. If V O = { u } = u + { 0 } then dim ⌈ E O ⌉ = dim { 0 } = 0 = lo g 2 | V O | ; if V O = { u , v } = u + { 0 , v − u } then dim ⌈ E O ⌉ = dim ⌈ v − u ⌉ = 1 = log 2 | V O | , hence according to Lemma 7 the constraint C is linear. ✸ This r esult holds b oth for the decis io n and the sear ch problem. 11 6 NP-Completeness results 6.1 Constrain ts with more than 2 elemen ts If one V O has more than 2 ele ments then C may not be a linear constra in t, and therefore the previous po ly nomial a lg orithm may not apply . In fact, w e now prove that allowing constraints of a car dinalit y greater than 2 makes the decision problem gc NP-hard, whatever the v alue of p . W e pr oceed by reduction from the problem o f 1-satisfia bilit y o f p ositive k -clauses. Let Σ b e a finite set of pr op ositional variables (which will b e denoted by Greek letters ), a p ositive k -clause is a subset C ⊆ Σ of cardina lit y k . Let S b e a finite set o f such clause s, then S is 1 -satisfiable if there is an interpr etation I ⊆ Σ such that every cla use C ∈ S co n tains exactly one element in I . Given Σ and S , we build an instance o f the decision pro blem gc (res tricted to F p -vector spaces) whose sa tisfiabilit y is equiv alen t to the 1-sa tisfia bilit y of S . F urthermore, the construction is p olynomial in the size o f the input cla use set, i.e., it is p olynomial in | Σ | and | S | (not nece s sarily in k , which is a constant). This transfor mation consis ts in in terpreting pro positiona l v ariables and clauses in F p . W e cons ider the spa ce of functions from Σ to F p , written F Σ p , with the standard s um ( u + v )( α ) = u ( α )+ v ( α ) and t he external pro duct ( xu )( α ) = xu ( α ) for all α ∈ Σ, u, v ∈ F Σ p and x ∈ F p . This is an F p -vector space of dimension | Σ | , with genera ting s e t { δ β | β ∈ Σ } , where δ β ( α ) = 1 if α = β and 0 otherwise. W e cons tr uct a p ermutation group as a homomo rphic imag e of the elemen- tary Ab elian p -group F Σ p . The elements to b e p ermuted are thos e of the set A S = ] C ∈ S F C p of functions fr o m C to F p , where C is a p ositive k -claus e b elonging to S . The cardinality of A S is P C ∈ S p | C | = p k | S | . Each ele ment of A S is a function that ass ociates integers mo dulo p to k prop ositional v ariables, hence it c an b e enco ded in cons tan t size. Example. W e consider the set Σ = { α, β , γ } a nd assume that p = 2. The r e are exactly 8 functions fr om Σ to F 2 , which we denote according to the following scheme. Σ g 0 g 1 g 2 g 3 g 4 g 5 g 6 g 7 α 0 0 0 0 1 1 1 1 β 0 0 1 1 0 0 1 1 γ 0 1 0 1 0 1 0 1 Let C = { α, β , γ } , which is a p ositive 3-clause on Σ, and S = { C } . Since C = Σ, we have A S = ] C ′ ∈ S F C ′ 2 = F C 2 = F Σ 2 = { g 0 , . . . , g 7 } . ✸ 12 Let f deno te the function from F Σ p to Sym( A S ) defined for all u ∈ F Σ p , C ∈ S and w ∈ F C p , by w f ( u ) = w + u | C . It is straig h tforward to verify that f ( u ) is indeed a p erm utation of A S , and that f ( u ) f ( v ) = f ( u + v ); hence f is a g r oup morphism and the gr o up G S generated by the p erm utations { f ( δ α ) | α ∈ Σ } is an elemen tary Ab elian p -g r oup. This generating set can obviously be computed in time p olynomial in | Σ | and | A S | . Example. W e compute f ( g 3 ). F or a ll w ∈ F Σ 2 , w f ( g 3 ) = w + g 3 . W e hav e g 0 + g 3 = g 3 , g 1 + g 3 = g 2 , etc. and we ea sily obtain, in cycle notation f ( g 3 ) = ( g 0 g 3 )( g 1 g 2 )( g 4 g 7 )( g 5 g 6 ) . ✸ Finally , let C S be the k -co nstrain t on G S defined for a ll C ∈ S and w ∈ F C p , by C S ( w ) = { w + ( δ α ) | C | α ∈ C } . Example. Since δ C ( α ) = g 4 , δ C ( β ) = g 2 and δ C ( γ ) = g 1 , we hav e for a ll w ∈ F C 2 , C S ( w ) = { w + g 4 , w + g 2 , w + g 1 } . This yields for instance C S ( g 3 ) = { g 7 , g 1 , g 2 } . ✸ Lemma 10 S is 1-satisfiable iff C S is satisfiable in G S . Pr o of. First assume that S is 1-sa tisfiable, i.e., there is a n interpretation I ⊆ Σ such that every cla use C ∈ S has exactly one elemen t in I . Let u ∈ F Σ p be defined by u ( α ) = 1 if α ∈ I and 0 o therwise. F or C ∈ S , if { α } = C ∩ I then it is clear that u | C = ( δ α ) | C , so tha t for all w ∈ F C p , w f ( u ) = w + u | C = w + ( δ α ) | C ∈ C S ( w ). Thu s f ( u ) ∈ G S satisfies C S . Conv ersely , supp ose there is an element f ( u ) of G S that satisfies C S , let I = { α ∈ Σ | u ( α ) = 1 } and let C b e a n y c la use in S . F or a ll w ∈ F C p , since w f ( u ) ∈ C S ( w ) ther e is an α ∈ C such that w f ( u ) = w + ( δ α ) | C , hence such that u | C = ( δ α ) | C . Necess arily u ( α ) = 1 and therefore α ∈ C ∩ I . If β ∈ C ∩ I we similarly o btain u | C = ( δ β ) | C , hence β = α . This prov es that C ∩ I is a singleton for all C ∈ S , and tha t I 1-sa tisfies S . ✸ Theorem 11 F or any prime p , the pr oblem of solving k -c onstr aints in F p -ve ctor sp ac es is N P-c omplete if k ≥ 3 . This follows from the NP- completeness o f the pr oblem of determining 1 - satisfiability of a set of p ositive 3-cla uses (see [4, pr o blem L04, p. 25 9 ]). 6.2 Constrain ts with at most 2 elemen ts If a V O has exa c tly t wo elements but p ≥ 3, then V O cannot b e a linear v ariety of F a nd the algo rithm of Section 5 .2 necessa rily fails. W e prove that allowing p ≥ 3 makes the restrictio n of the decisio n problem gc to F p -vector spaces 13 NP-complete, ev en with constraints of ca rdinalit y at mo st 2 . W e proceed by reduction fro m 1-satisfia bilit y o f p o sitiv e p -cla us es. Given a se t S o f p ositiv e p - clauses on Σ, we again consider the F p -vector space F Σ p and define the se t A ′ S = Σ × F p ⊎ S × F p , whose c ardinality is p | Σ | + p | S | . Let f ′ be the function from F Σ p to Sym( A ′ S ) defined for all u ∈ F Σ p , h α, x i ∈ Σ × F p and h C , y i ∈ S × F p , by h α, x i f ′ ( u ) = h α, x + u ( α ) i , h C, y i f ′ ( u ) = h C, y + X β ∈ C u ( β ) i . It is straightforward to verify that f ′ ( u ) is a p ermutation of A ′ S : if h α, x i f ′ ( u ) = h α ′ , x ′ i f ′ ( u ) then α = α ′ and then x = x ′ ; if h C, y i f ′ ( u ) = h C ′ , y ′ i f ′ ( u ) then C = C ′ and then y = y ′ . It is obvious that f ′ ( u ) f ′ ( v ) = f ′ ( u + v ), hence f ′ is a gro up morphism and the g roup G ′ S generated by the p erm utations { f ′ ( δ α ) | α ∈ Σ } is therefore an elementary Abelian p -g roup. This g e nerating set can b e computed in time p olynomial in | Σ | and | A ′ S | . Example. W e assume th e same Σ, p , C and S a s in the running example of Section 6.1. Then A ′ S = {h α, 0 i , h α, 1 i , h β , 0 i , h β , 1 i , h γ , 0 i , h γ , 1 i , h C, 0 i , h C, 1 i} . The pe rm utations f ′ ( u ) for u ∈ F Σ 2 can a gain b e express e d in cycle nota tion, for instance: f ′ ( g 2 ) = ( h β , 0 i h β , 1 i )( h C, 0 i h C , 1 i ) , f ′ ( g 3 ) = ( h β , 0 i h β , 1 i )( h γ , 0 i h γ , 1 i ) . h C, 0 i is a fix-p oint of f ′ ( g 3 ) since g 3 ( α ) + g 3 ( β ) + g 3 ( γ ) = 0 + 1 + 1 = 0. ✸ Let C ′ S be the 2-cons train t on G ′ S defined, for all h α, x i ∈ Σ × F p and h C, y i ∈ S × F p , by C ′ S ( h α, x i ) = {h α, x i , h α, x + 1 i} , C ′ S ( h C, y i ) = {h C, y + 1 i} . Example. Obviously C ′ S ( h α, 0 i ) = C ′ S ( h α, 1 i ) = {h α, 0 i , h α, 1 i} , and similarly for β and γ . F or the other tw o elements of A ′ S we have: C ′ S ( h C, 0 i ) = {h C , 1 i} and C ′ S ( h C, 1 i ) = {h C , 0 i} . ✸ Lemma 12 S is 1-satisfiable iff C ′ S is s atisfi able in G ′ S . 14 Pr o of. Assume S is 1-s atisfiable, le t I b e a n int erpreta tio n of S and co nsider u ∈ F Σ p defined by u ( α ) = 1 if α ∈ I and 0 otherwise. The co nstrain t on any h α, x i ∈ Σ × F p is satisfied since h α, x i f ′ ( u ) = h α, x + u ( α ) i ∈ C ′ S ( h α, x i ). F or all h C , y i ∈ S × F p there is a unique β ∈ C s uc h tha t u ( β ) = 1 and u | C is zero elsewhere, hence h C , y i f ′ ( u ) = h C, y + P β ∈ C u ( β ) i = h C, y + 1 i ∈ C ′ S ( h C, y i ). This shows that f ′ ( u ) ∈ G ′ S satisfies C ′ S . Conv ersely , suppos e that an element f ′ ( u ) of G ′ S satisfies C ′ S and let I = { α ∈ Σ | u ( α ) = 1 } . Then u ( α ) 6 = 1 for all α ∈ Σ \ I , and u ( α ) ∈ { 0 , 1 } since h α, x i f ′ ( u ) ∈ C ′ S ( h α, x i ), thus u ( α ) = 0 (mo dulo p ). Let C be a clause in S , the constraint yields h C, 0 i f ′ ( u ) = h C , 1 i , hence P β ∈ C u ( β ) = 1. The terms of this sum a r e either 0 or 1 , hence at leas t one m ust b e a 1. F urthermore, there are at most p terms, hence o nly one can be a 1, say u ( α ) = 1 , then b y defi- nition of u , α is the only mem ber of C tha t b elongs to I . Hence I 1-satisfies S . ✸ Theorem 13 F or any prime p ≥ 3 , the pr oblem of solving 2-c onst ra ints in F p -ve ctor sp ac es is NP-c omplete. 7 Exp erimen tal results The p olynomial algo r ithm for solving linear constraints has b een implemented in the GAP sy s tem, using its facilities on p erm utations, matrix alg ebra and finite fields . The implementation, nicknamed Sol vect (see “do wnloads” page on capp.im ag.fr ), is straig h tforward except for the fact that co ordina tes in the trans itiv e constituents a re kept in memor y a nd hence co mputed only once (this is p erformed while co mputing the orbit pa rtition o f G ). Its p erformance has bee n measured ag ainst a g eneral purp ose gro up search a lg orithm provided in GAP , descr ib ed in [6] and refined in [11]. The call to this algo rithm is Elemen tPropert y ( ⌈ g 1 , . . . , g m ⌉ , g 7→ ∀ a ∈ A, a g ∈ C ( a ) ); which returns a n element of G = ⌈ g 1 , . . . , g m ⌉ satisfying the sp ecified prop erty if there is o ne, and fail o therwise. The p erformance of E lementPr operty dep ends esse n tially on the size of G , while Solv ect dep ends mostly on n = | A | and to a lesser extent on d = dim F . W e thus p erform tw o sets of exp erimen ts: the first in T able 1 is parametriz ed by the size of G (whic h is 2 dim G ) a nd the second in T able 2 is parametr ized by n . In each cas e we measure the mean v alues o f n , dim G a nd d a s well as the times in milliseconds taken by the tw o solvers ( t 1 for S olvect and t 2 for Elemen tPropert y ). The samples are gener ated by choos ing ra ndo mly the n um b er and dimensions of tra nsitiv e constituents, i.e ., a sequence d 1 , . . . , d q of strictly p ositive integers, then computing gener ators for the tra nsitiv e constituents and co mposing them randomly to pro duce generators for G . In the first exp eriment w e guar an tee that G has the corr ect dimension, bounded by max q i =1 d i ≤ dim G ≤ P q i =1 d i = d . 15 dim G n d t 1 t 2 5 26 . 4 ± 56% 6 . 9 ± 29% 0 . 38 4 ± 310% 0 . 82 ± 214% 10 270 ± 150% 14 . 9 ± 26 % 1 . 46 ± 137% 2 0 . 3 ± 56% 15 785 ± 277% 27 . 2 ± 28 % 5 . 4 ± 143 % 631 ± 106% 20 1 060 ± 229 % 35 . 4 ± 28% 10 . 1 ± 100 % 19 90 0 ± 90% 25 2 230 ± 175 % 52 . 2 ± 31% 25 . 8 ± 73% − 30 2 730 ± 148 % 67 . 4 ± 34% 45 . 3 ± 66% − 35 2 870 ± 107 % 79 ± 35% 65 . 3 ± 67 % − 40 8 510 ± 94% 9 4 . 2 ± 38% 14 7 ± 69 % − 45 7 680 ± 77% 125 ± 37% 2 41 ± 64% − 50 12 8 00 ± 60% 14 7 ± 39% 436 ± 75% − T able 1: Exp e rimen t 1 In the se cond exp erimen t we g ua rant ee tha t P q i =1 2 d i = n . 2-constra in ts are also ge nerated rando mly , with the following bia s : half of them a re guar an teed to b e satis fia ble (a s o lution is chosen r andomly in G ). Another bias has b een int ro duced: we cho ose the d i betw een 1 and 13, b ecause c o mputing genera tors for an orbit of a s iz e greater than 2 13 takes too muc h time. Since our random samples are by no means supp osed to b e representative, we also mea s ure the standard deviation expre s sed as a p ercentage of the mean v alue 3 . W e test 100 0 ins tances o n the low v alues and 10 0 on the higher ones. V alues are rounded to 3 digits. W e see tha t E lementPro perty can har dly be used on gro ups of size muc h bigger than 2 20 , while Solve ct works well up to the limits of the memory used by GAP (the limit is r each ed with n = 2 17 ). 8 Conclusion and p ersp ectiv es W e can therefore solve k -constr ain ts in the c la ss o f F p -vector s paces in guar an- teed poly nomial time only when k = p = 2 , and we have pro vided a n alg orithm to do s o. F or gr eater v alues of k and p the problem is NP-co mplete, which is quite sur prising considering the rich structure of vector spa ces a nd the r elativ e simplicity of the constraints that were cons ide r ed. Thes e res ults confirm how difficult it can b e to develop efficient a lgorithms for finding useful symmetries. How ev er, our algo r ithm may b e used on linear constraints rega r dless o f k and p , and o ther exp eriments with Solv ect suggests that many co nstraint s are linear. F urthermor e, chec king the linearity of the constra in t is fast. But this still requires the gr oup to be an elementary Ab elian p -group, which seems unlikely unless the problem under consideration is of a geometric na ture, for instance if a h yp ercube is in volv ed (its gr oup o f symmetries is an elemen tary 3 When a pro cess takes less than 4 ms, GAP measures its du ration as either 0 or 4 ms, hopefull y with a probabilit y depending on this duration. In that case the mean v alue should be accurate, but standard deviation is obviously exaggerated. 16 n dim G d t 1 t 2 2 1 1 1 0 . 108 ± 600% 0 . 28 ± 375% 2 2 1 . 81 ± 21% 2 0 . 144 ± 517% 0 . 308 ± 351 % 2 3 2 . 96 ± 20% 3 . 6 8 ± 13% 0 . 212 ± 431% 0 . 44 4 ± 294% 2 4 4 . 44 ± 19% 6 . 3 9 ± 22% 0 . 344 ± 342% 0 . 75 6 ± 211% 2 5 6 . 12 ± 19% 10 . 4 ± 30% 0 . 716 ± 224% 1 . 88 ± 13 9% 2 6 8 . 05 ± 19% 16 . 2 ± 35% 1 . 4 ± 141% 7 . 05 ± 145% 2 7 10 ± 19% 2 4 . 4 ± 36% 3 . 08 ± 92% 35 . 8 ± 188% 2 8 12 ± 19% 3 4 . 1 ± 40% 6 . 96 ± 79% 1 84 ± 277% 2 9 14 . 5 ± 18% 51 . 1 ± 33% 17 . 5 ± 74% 1 740 ± 322% 2 10 16 . 6 ± 17% 67 . 6 ± 34% 34 . 9 ± 81% 1 5 7 00 ± 426% 2 11 18 . 5 ± 17% 85 . 2 ± 37% 64 . 4 ± 69% 3 7 6 00 ± 166% 2 12 20 . 3 ± 15% 113 ± 37% 1 44 ± 88% 187 000 ± 315% 2 13 23 . 3 ± 12% 131 ± 32% 2 09 ± 66% − 2 14 25 . 8 ± 9% 189 ± 26% 593 ± 80 % − 2 15 28 . 8 ± 8% 268 ± 23% 1 520 ± 60% − 2 16 30 . 9 ± 7% 446 ± 17% 6 910 ± 55% − T able 2: Expe r imen t 2 Abelia n 2- group). In gener al it would b e necessary to enforce this pr o perty by approximating the group of s y mmetries by one or several elementary Ab elian p -subgro ups. This is reasonable since s ymmetry pruning is mea nt to b e fas t, not complete with resp ect to just any group of symmetries. It seems p ossible to generalize these results to the class of Ab elian g roups in the line of [7], a t the exp ense of our elegant geometric in terpretation o f linear constraints. W e are currently inv estigating this genera lization. Ident ifying tractable restr ictions of the generally intractable problem of find- ing selected s ymmetries is t herefore a natur al approa c h to efficien t s ymmetry pruning. Metho ds using only sp ecial symmetries hav e already b e en tried with some s uccess, as in [9] where only transp ositions a re consider ed. W e therefore belie v e the pr esen t results op en interesting pe r spectives. References [1] Butler, G.: F undamental algorithms for p ermutation groups. Lecture Notes in Computer Science 559, Springe r V erlag (1 9 91) [2] Crawford, J.M., Ginsber g, M.L., Luks, E .M., Roy , A.: Symmetry-break ing predicates fo r sear c h pr oblems. In: KR. pp. 14 8–159 (19 96) [3] Dummit, D.S., F o ote, R.M.: Abstract Algebra. John Wiley and Sons (199 9) 17 [4] Garey , M., Johnso n, D.S.: Computers and intractability: a guide to the theory o f NP -completeness. F reeman, San F r ancisco, California (1979) [5] Geddes, K.O., Cza por, S.R., L abahn, G.: Algo r ithms for Computer Alge- bra. Kluwer Academic Publishers Group (1 992) [6] Leon, J.S.: Permutation gr oup algor ithms base d on par titions, I: Theor y and a lgorithms. Journal of Symbolic Co mputation 12, 53 3–583 (1991) [7] Luks, E.M., Roy , A.: The co mplex it y of symmetry- breaking formulas. An- nals of Mathematics and Artificia l In telligence 41 (1), 19–45 (200 4 ) [8] McKay , B.: Naut y user s guide (version 1 . 5). T echnical rep ort, Dept. Co mp. Sci., Austra lian National University (19 90) [9] Peltier, N.: A new metho d for auto mated finite mo del building exploiting failures and symmetries. Journa l of Logic and Computation 8(4), 511–5 43 (1998) [10] Sco tt, W.R.: Group Theo ry . Dov er, New Y ork (19 87) [11] Theiß en, H.: Eine Metho de zur Normalisa torber ec hn ung in Permutations- grupp en mit An wendungen in de r Konstr uktion primitiver Gruppen. Ph.D. thesis, R WTH Aa c hen, Germany (1997) [12] W alsh, T.: Parameter iz e d complexity r esults in symmetry breaking . CoRR abs/10 09.1174 (201 0), http ://arxiv .org/abs/1009.1174 , informal pub- lication 18