On the Power of Symmetric Linear Programs
aa r X i v : . [ c s . L O ] J a n On the Power of Symmetric Linear Programs
Albert Atserias Anuj Dawar Joanna Ochremiak Universitat Polit`ecnica de Catalunya University of Cambridge January 24, 2019
Abstract
We consider families of symmetric linear programs (LPs) that decide a property ofgraphs (or other relational structures) in the sense that, for each size of graph, thereis an LP defining a polyhedral lift that separates the integer points corresponding tographs with the property from those corresponding to graphs without the property.We show that this is equivalent, with at most polynomial blow-up in size, to familiesof symmetric Boolean circuits with threshold gates. In particular, when we considerpolynomial-size LPs, the model is equivalent to definability in a non-uniform version offixed-point logic with counting (FPC). Known upper and lower bounds for FPC applyto the non-uniform version. In particular, this implies that the class of graphs withperfect matchings has polynomial-size symmetric LPs while we obtain an exponentiallower bound for symmetric LPs for the class of Hamiltonian graphs. We compare andcontrast this with previous results (Yannakakis 1991) showing that any symmetric LPsfor the matching and TSP polytopes have exponential size. As an application, weestablish that for random, uniformly distributed graphs, polynomial-size symmetricLPs are as powerful as general Boolean circuits. We illustrate the effect of this on thewell-studied planted-clique problem. Introduction
The theory of linear programming is a powerful and widely-used tool for studying combi-natorial optimization problems. By the same token, the limitations of such methods are animportant object of study in complexity theory. A major step in this line of work was theseminal paper of Yannakakis [40] that initiated the study of symmetric linear programs forcombinatorial problems.A polytope in R n is the convex hull of a finite set of points in R n . Dually, it is theintersection of the finite number of half-spaces that define its facets. Consider a language S ⊆ { , } ∗ and let S n ⊆ { , } n be the collection of strings in S of length n . We canassociate with S n the polytope P n ⊆ R n that is the convex hull of the points x ∈ R n with 0-1 coordinates that correspond to the strings in S n . If this polytope has a succinctrepresentation as a system of linear inequalities, we can use linear programming methodsto optimize linear functions over S n . In general, a succinct representation might mean thatits size grows polynomially with n . Thus, the size of the polytope P n , say measured by thenumber of its facets, is an important measure of the complexity of S .In general, even when P n has a large number of facets, it may admit a succinct repre-sentation as the projection onto R n of a polytope Q ⊆ R n + m of higher dimension. In thissituation, we call Q a lift of P n and P n a shadow of Q . This is the basis for so-called extendedformulations of combinatorial optimization problems. It allows us to optimize over S n usinglinear programs with auxiliary variables. A classic example is the convex hull of all stringsin { , } n of odd Hamming weight, known as the parity polytope which has exponentiallymany facets but has an extended formulation using only polynomially many inequalities.An interesting feature of many such examples of small extended formulations is that theyare strongly symmetric, i.e., any basic automorphism of the shadow polytope extends to anautomorphism of its lift.Yannakakis [40] established lower bounds on the size of symmetric lifts for the perfectmatching polytope and the travelling salesman polytope. The perfect matching polytope on2 n vertices is the convex hull of points in { , } E where E = (cid:0) [2 n ]2 (cid:1) which represent the edgesets of a perfect matching on 2 n vertices. Yannakakis shows that any symmetric lift Q ofthis polytope necessarily has a number of facets that is exponential in n . Here “symmetric”means that any permutation of the n vertices extends to an automorphism of Q . This lowerbound is then used to show a similar lower bound for symmetric lifts of the Hamilton cyclepolytope (also known as the travelling salesman polytope). This is the convex hull of pointsin { , } E where E = (cid:0) [ n ]2 (cid:1) which are the edge sets of Hamilton cycles of length n . Theconclusion is that any attempt to solve the travelling salesman problem by representingit as a linear program in a natural way (i.e. respecting the symmetries of the graph) isdoomed to be exponential. These results launched a long study of extended formulations ofcombinatorial problems. Relatively recently, exponential lower bounds have been establishedeven without the assumption of symmetry [35].There is another way of representing a language S ⊆ { , } ∗ by a family of polytopesthat is also considered by Yannakakis. Say that S n is recognized by a polytope P n if S n ⊆ P n and { , } n \ S n is disjoint from P n . In particular, the convex hull of S n recognizes S n , but2t may well be that there are more succinct polytopes that also do. Indeed, Yannakakisshows that for any language S decidable in polynomial time, there is a family of polynomial-size polytopes whose shadows recognize S n . Thus, we cannot expect to prove exponentiallower bounds on such polytopes without separating P from NP. Note that the assumptionof symmetry has been dropped here. What can we say about symmetric lifts of polytopesrecognizing S n ? Yannakakis does not consider this question and it does not appear to havebeen studied in the literature. This is the question that we take up in this paper.We consider families of symmetric polytopes for recognizing classes of graphs (or otherrelational structures). This gives an interesting contrast with the results of Yannakakis. Ourresults show that there is a polynomial-size family of symmetric polytopes whose shadowsrecognize the class of graphs that contain a perfect matching. On the other hand, there isno family of symmetric polytopes of sub-exponential size whose shadows recognize the classof graphs with a Hamiltonian cycle.We obtain these specific upper and lower bounds by relating the power of symmetric linearprograms to two other natural models of symmetric computation, based respectively on logicand circuits. To be precise, we show that families of symmetric polytope lifts for recognizinga class of structures are equivalent to families of symmetric Boolean circuits with thresholdgates, in the sense that there are translations between them with at most a polynomial blow-up in size in either direction. This places symmetric linear programs squarely in the contextof a fairly robust notion of symmetric computation that has recently emerged. It was shownin [3] that P-uniform families of symmetric circuits with threshold gates are equivalent tofixed-point logic with counting (FPC), a well-studied logic in descriptive complexity theory(see [14]).Our translation from circuits to linear programs is based on that given by Yannakakis,but we need to preserve symmetry and, for threshold gates, this poses a significant challenge.To construct symmetric linear programs that enforce the values of threshold gates we needa sweeping generalization of the construction of symmetric lifts of the parity polytope. Inthe other direction, we make a detour through logic. That is, we show how a family ofsymmetric polytopes can be translated into a family of formulas of first-order logic withcounting, with the number of variables and the size being tightly bounded based on thesize of the polytopes. The translation is based on a support theorem, which allows us tointerpret in the logic, given a linear program P as advice, a version of P for a particularinput structure. This then allows us to use the result of [4] to the effect that solvability oflinear programs is definable in FPC.It is interesting to compare our results with the equivalence between FPC and P-uniformsymmetric threshold circuits established in [3]. Our results are stated for the non-uniformmodel and it is not clear that they can be made uniform. In particular, our translationfrom linear programs to formulas of counting logic, while it preserves size, is not necessarilycomputable in polynomial time. It involves symmetry checks that are as hard as the graphisomorphism problem. On the other hand, the results in [3] were stated for polynomial-sizefamilies of circuits and we are able to extend them to sizes up to weakly exponential. Thetranslation from circuits to formulas given in [3] was based on a support theorem proved there3hich only worked for circuit sizes bounded by O (2 n / ). We use a stronger support theorem(proved in Section 4.2) which enables us to prove the translation from families of symmetriclinear programs to formulas of counting logic for sizes up to O (2 n − ǫ ) for arbitrarily small ǫ .The upper and lower bounds for symmetric linear programs that we obtain (such as forthe perfect matching and the Hamilton cycle problem, respectively) are direct consequencesof the equivalence with non-uniform counting logic. For instance, it is known [4] that perfectmatching is definable in FPC and it follows that it is recognized by a polynomial-size family ofsymmetric polytope lifts. Inexpressibility results for FPC are usually established by showinglower bounds on the number of variables required to express a property in counting logic,and they yield lower bounds even in the non-uniform setting. In particular, we tightenknown lower bounds on Hamiltonicity to show that it cannot be expressed with a sub-linearnumber of variables hence with weakly exponential size symmetric polytope lifts. Indeed,if we use the strongest form of our translation from symmetric LPs to logic formulas, thenthe lower bound we get for Hamiltonicity is even exponential, i.e., of type 2 Ω( n ) where n is the number of vertices of the graph. Similar exponential lower bounds for other NP-complete problems (such as graph 3-colourability and Boolean satisfiability) follow fromknown bounds in counting logic. Indeed, exponential lower bounds for some problems inP (such as solving systems of linear equations over finite fields) also follow. It should benoted that this establishes exponential lower bounds also on symmetric threshold circuits, aproblem left open in [3], where superpolynomial lower bounds were established.Another consequence can be derived from the connection with FPC. We know thatFPC can express all polynomial-time properties of almost all structures under a uniformdistribution (see [23]). This can be used to show that FPC can solve the planted cliqueproblem if, and only if, the problem is solvable in polynomial time. The planted cliqueproblem is that of distinguishing a random graph from one in which a clique has beenplanted. It is a widely studied problem in the context of lower bounds on linear programmingmethods (see e.g. [1, 19, 7, 26]). It is a consequence of our results that if this problem can besolved in polynomial time, then it is solvable by polynomial sized symmetric linear programs.This is significant because a number of lower bounds have been established for the plantedclique problem for a variety of models of linear and semidefinite programming, notably thewell-studied Lov´asz-Schrijver, Sherali-Adams and Lasserre hierarchies. It is noteworthy thatall of these hierarchies yield symmetric linear or semidefinite programs. Our results showthat these lower bounds cannot be extended to general symmetric linear programs withoutseparating P from NP.In Section 2 we establish some preliminary definitions and notation. Section 3 givesthe translation of circuits to linear programs. This translation is carried out for a verygeneral notion of symmetry. For the reverse translation, from linear programs to logic givenin Section 4, we restrict to the natural symmetries on graphs and relational structures.The main result and its consequences, including upper and lower bounds are presented inSection 5. 4 Preliminaries
For a natural number n ∈ N , we write [ n ] for { , . . . , n } with the understanding that [0] = ∅ .For any set X , by Sym X we denote the symmetric group on X , that is, the group of allpermutations of the set X , and by Alt X we denote the alternating group on X , that is, thegroup of all even permutations of the set X . In the special case of X = [ n ] we write Sym n and Alt n , respectively. Logarithms are base 2 with the convention that log(0) = 0. By H ≤ G we denote the fact that H is a subgroup of G . If H ≤ G then, for any g ∈ G ,the subset gH = { gh : h ∈ H } of G is called a coset of H in G . The number of such cosetsis called the index of H in G and is denoted by [ G : H ].Recall that for any group G , a G - set is a set X with an action of the group G , where byan action we mean a mapping · : G × X → X such that for any π, σ ∈ G and any x ∈ X we have π · ( σ · x ) = πσ · x and id · x = x . Equivalently, X is a G -set if it comes with ahomomorphism from the group G to the symmetric group Sym X on X . A homomorphism from a G -set X to a G -set Y is a function f from X to Y such that for any π ∈ G and x ∈ X , it holds that π · f ( x ) = f ( π · x ). For a G -set X , a stabilizer of an element x ∈ X in G consists of all π ∈ G such that π · x = x . It is easy to see that a stabilizer is a subgroupof G . We sometimes denote it by G x .For any set X , by | X | we denote the number of elements in X , by P ( X ) we denote thepower set of X and by X n we denote the set of n -tuples of elements of X . Moreover, if n ≤ | X | , by X ( n ) we denote the set of all n -tuples of distinct elements of X . In particular,for n = 0, both X n and X ( n ) are one-element sets consisting of the empty tuple. If X is a G -set, then the action of G on X induces an action of G on each of the sets P ( X ), X n and X ( n ) in the natural way: for any π ∈ G , we have π · T = { π · x : x ∈ T } , where T ⊆ X , and π · s = ( π · s , . . . , π · s n ), where s = ( s , . . . , s n ) is a tuple from X n or X ( n ) . We refer to thelatter group action as the componentwise action of the group G .An action of a group G on a set of indices U defines an action of G on the set of indexedvariables { x u } u ∈ U in the natural way: π · x u = x π · u , for any π ∈ G and u ∈ U . This extendsto vectors of indexed variables, as discussed in the paragraph above. For instance, if theset of indices [ n ] comes with the componentwise action of the group Sym n , then for any π ∈ Sym n and x = ( x ij ) i,j ∈ [ n ] , we have π · x = ( x π ( i ) π ( j ) ) i,j ∈ [ n ] . From now on, in the case ofvectors of variables we use the notation x π instead of π · x .For any G -set U , we define an action of G on the real vector space R U in the followingway. First, the action of G on the standard basis { e u } u ∈ U , where e u is the vector whose u -th coordinate is 1 and all other coordinates are 0, is given by π · e u = e π · u , for any π ∈ G and u ∈ U . This way each π ∈ G defines a mapping from { e u } u ∈ U to { e u } u ∈ U . Theaction of G on the vector space spanned by { e u } u ∈ U can be seen as the linear extensionof those mappings: for any π ∈ G and any real vector a = P u ∈ U a u e u , we have π · a = P u ∈ U a u ( π · e u ) = P u ∈ U a u e π · u . For instance, if the set of indices [ n ] comes with the naturalaction of the group Sym n , then for any π ∈ Sym n and for any vector a = ( a , . . . , a n ), we5ave π · a = P i ∈ [ n ] a i e π ( i ) = P i ∈ [ n ] a π − ( i ) e i = ( a π − ( i ) , . . . , a π − ( n ) ). Here again we use thenotation a π instead of π · a . This notational convention extends to subsets of real vectorspaces: for P ⊆ R U we write P π instead of π · P .If a group G acts on a set U and a group H acts on a set W , then the product group G × H acts on the disjoint union U . ∪ W : given π ∈ G and σ ∈ H , we have ( π, σ ) · u = π · u ,for u ∈ U , and ( π, σ ) · w = σ · w , for w ∈ W . Given such an action of the product group G × H , of particular interest to us is its induced action on R U × R W and on sets of variablesindexed by U . ∪ W . A (many-sorted relational) vocabulary consists of a finite set of sort symbols and a finiteset of relation symbols. Each relation symbol R comes with an associated natural numberar( R ) called its arity and with an associated type which is a product of ar( R )-many sortsymbols U i × . . . × U i ar( R ) . The vocabulary L G of (directed) graphs is single-sorted and hasone relation symbol E of arity two. If L is a vocabulary, then an L -structure A is givenby disjoint sets U , . . . , U s , called domains , one for each sort symbol in L , and a relation R A ⊆ U i × . . . × U i r for each R ∈ L of arity r and type U i × . . . × U i r . The relation R A is called the interpretation of R in A . Whenever this does not lead to confusion we use U to denote the domain associated to the sort symbol U . Moreover, when A is clear from thecontext, we omit the superscript in R A . All our structures are finite: their domains are finitesets. A directed graph is an L G -structure; the graph is undirected if its interpretation of E is symmetric and irreflexive.In a logic for a many-sorted vocabulary L the variables are typed, that is, each differentsort has its own set of individual variables. When an L -formula is interpreted on an L -structure, the variables range over the domain of their sort. The atomic L -formulas areequalities between variables of the same type, and formulas of the form R ( x , . . . , x r ), where R is a relation symbol of arity r in L , and x , . . . , x r are variables of appropriate types.The class of formulas of first-order logic (FO) is the smallest class of formulas that containsall atomic formulas and is closed under negation, conjunction, and existential quantification.We consider an extension of first-order logic with counting quantifiers . For each naturalnumber q , we have a quantifier ∃ ≥ q where A | = ∃ ≥ q x φ if, and only if, there are at least q distinct elements a ∈ A such that A | = φ [ a/x ]. While the extension of first-order logic withcounting quantifiers is no more expressive than FO itself, the presence of these quantifiersdoes affect the number of variables that are necessary to express a query. Let C k denote the k -variable fragment of this logic, i.e. those formulas in which no more than k variables appear,free or bound. FPC is the extension of first-order logic with fixed-points and counting. Wedo not give a full definition here as it can be found in standard texts such as [33]. We notethat formulas of FPC have two sorts of variables, ranging respectively over the elements ofthe domain of interpretation and over natural numbers (restricted to the size of the domain),and allow for terms of the form xφ which denotes the number of elements that satisfy φ ( x ).We write FOC to denote the fragment of FPC without fixed-point operators, but where wedo allow arithmietic operations (+ and × ) on the number sort. The size of a formula is6efined as the number of its subformulas. For each formula φ of FPC (and, per force, FOC),if the formula uses k variables then, for every n , there is a formula θ n of C k such that φ isequivalent to θ n on all structures of size at most n . Moreover, θ n has size that is polynomialin the size of φ , in k , and in n k . For more on FPC and its relation to the bounded variablefragments C k we refer to [33].Rational numbers q ∈ Q are represented by structures of a single-sorted vocabulary L Q with three monadic relation symbols and one binary relation symbol ≤ . If q = ( − b n/d ,where n, d ∈ N and b ∈ { , } , then the domain of an L Q -structure that represents q is { , . . . , N } where N ∈ N is large enough to represent both the numerator and denominatorwith N bits. The binary relation ≤ is interpreted by the natural linear order on { , . . . , N } .The first of the monadic relation symbols of L Q is used to represent the sign b of q by havingit empty if, and only if, b = 0. The other two monadic relation symbols of L Q are used torepresent the bit positions on which the numerator n and the denominator d have a one. Weuse zero denominator to represent ±∞ .If I , . . . , I d denote index sets, tensors u ∈ Q I ×···× I d are represented by many-sorted struc-tures, with one sort ¯ I for each index set I on the list I , . . . , I d , and one sort ¯ B for a domain { , . . . , N } of bit positions. The vocabulary L vec ,d of these structures has a binary relationsymbol ≤ for the natural linear order on { , . . . , N } and three d + 1-ary relation symbols P s , P n and P d for encoding the signs and the bits of the numerators and the denominators ofthe entries of the tensor. Matrices A ∈ Q I × J and vectors a ∈ Q I are special cases of these.Indexed sets of vectors { a i : i ∈ K } ⊆ Q I and index sets of rationals { b i : i ∈ K } ⊆ Q too. A polytope is a set of the form P = { x ∈ R U : Ax ≤ b } , where U and V are abstractnon-empty index sets, A ∈ R V × U is a constraint matrix, and b ∈ R V is an offset vector. Ifwe think of x = ( x u ) u ∈ U as a sequence of variables, then the defining system of inequalityconstraints Ax ≤ b is called a linear program (LP) for P . Note that the defining LPsfor polytopes are by no means unique. Typically A and b can be chosen to have rationalentries, in which case P is represented by a sequence of linear constraints ( γ v ) v ∈ V of one ofits defining LPs with rational entries; i.e., each γ v is of the form a T v x ≤ b v , with a v ∈ Q U and b v ∈ Q . The size of such an LP is ( | U | + 1) | V | b , where b is the maximum number of bitsit takes to write all the numerators and all the denominators of the entries of the a v and b v in binary.If x ∈ R U and y , . . . , y m ∈ R U , and α , . . . , α m ∈ R are such that x = P mi =1 α i y i ,with α i ≥ P mi =1 α i = 1, then we say that x is a convex combination of y , . . . , y m .When 0 < α i < i ∈ [ m ], the convex combination is called non-trivial. The convexhull conv( y , . . . , y m ) is the set of all convex combinations of y , . . . , y m . A point x of apolytope P is called a vertex if it cannot be expressed as a non-trivial convex combinationof any two other points of P . If P is a polytope and P ⊆ { x ∈ R U : a T x ≤ b } , thenthe set { x ∈ P : a T x = b } is called face of P . The faces of dimension 0 are the verticesof P ; the faces of dimension 1 are called edges ; the faces of dimension dim( P ) − facets , where dim( P ) is the dimension of P . Each polytope has finitely many faces of each7imension; in particular finitely many vertices (see [36]). A polytope is bounded if and onlyif it is the convex hull of its vertices.If P ⊆ R U × R W is a polytope, its projection into R U is the set of points x ∈ R U forwhich there exists a point y ∈ R W with ( x , y ) ∈ P . The projection of a polytope is againa polytope. If Q is the projection of P into R U , then we say that Q is a shadow of P , andthat P is a lift of Q . If A, B ⊆ { , } U are disjoint, then we say that Q is a polytope thatseparates A from B if A ⊆ Q and B ⊆ R U \ Q . We also say that P is a polytope lift thatseparates A from B . If Q separates A from its complement A = { , } U \ A , then we saythat Q is a polytope that recognizes A , and that P is a polytope lift that recognizes A . Sinceno point in { , } U is in the convex hull of any set of points in { , } U that does not containit, the convex hull of A ⊆ { , } U always recognizes A . The converse is not true: a polytopecould recognize A and not be the convex hull of A .Let P ⊆ R U be given by a sequence of linear constraints ( γ v ) v ∈ V . If a group G acts on theset U , then for any γ v of the form a v x ≤ b v , we write γ πv for the linear constraint a v x π ≤ b v .Note that the sequence ( γ πv ) v ∈ V defines P π ⊆ R U , which is again a polytope. As long as thisdoes not lead to confusion, we identify polytopes with sequences of linear constraints thatrepresent them. In particular, if we assume the polytope P ⊆ R U to be represented by thesequence of constraints ( γ v ) v ∈ V , then by P π we mean both the permuted polytope itself, andits representation by the sequence of constraints ( γ πv ) v ∈ V .Let U be a G -set. A polytope P ⊆ R U × R W is said to be G - symmetric if for every π ∈ G there exists a permutation σ ∈ Sym W such that P ( π,σ ) = P . If additionally we aregiven an action of the group G on W such that P ( π,π ) = P , then we say that the polytope P is G -symmetric with respect to this action . A pair of permutations ( π, σ ) ∈ Sym U × Sym W such that P ( π,σ ) = P is called an automorphism of P . Hence, if G ≤ Sym U , the fact that thepolytope P is G -symmetric is equivalent to the possibility of extending every permutation π ∈ G to an automorphism of P .For any n ∈ N , if the set [ n ] comes with the natural action of the symmetric groupSym n , then any Sym n -symmetric polytope P ⊆ R [ n ] × R W is said to be graph-symmetric . Itis not difficult to see that any set A ⊆ { , } [ n ] recognised by a graph-symmetric polytopelift P ⊆ R [ n ] × R W is invariant with respect to the action of the group Sym n , i.e., for any a ∈ A and any π ∈ Sym n , we have a π ∈ A . The polytope lift P can be therefore seen asrecognising a class of graphs with n -vertices. If we take a graph G with the set of vertices V of size n , fix a bijection f from [ n ] to V , and define a vector a = ( a ij ) i,j ∈ [ n ] ∈ { , } [ n ] by: a ij = 1 if and only if there is an edge from f ( i ) to f ( j ) in G , then G belongs to the classrecognised by P if and only if a ∈ A . Since A is a Sym n -set, this definition does not dependon the choice of f .More generally, we consider Sym n -symmetric polytope lifts recognising properties of ar-bitrary L -structures. For any n ∈ N and any single-sorted vocabulary L , let L ( n ) be thedisjoint union of [ n ] ar( R ) over all relation symbols R in L . Since L ( n ) comes with the naturalaction of the group Sym( n ), we can talk about Sym n -symmetric polytopes over R L ( n ) × R W .Any such polytope is called L - symmetric . As a straightforward generalisation of the discus-sion in the previous paragraph, L -symmetric polytope lifts over R L ( n ) × R W are defined to8ecognise classes of L -structures with n -element domains. A circuit with inputs ( x u ) u ∈ U is a directed acyclic graph G whose vertices of zero in-degreeare labelled by some input x u , and every other vertex is labelled by a function from somefixed basis of symmetric Boolean functions, with the constraint that the function takesthe same number of inputs as the in-degree of the vertex. The vertices of zero out-degreeare called outputs. If ( y v ) v ∈ V is a fixed naming of the outputs, then a circuit computes aBoolean function f : { , } U → { , } V in the obvious way. When m = 1 we say that itrecognizes f − (1). The vertices of a circuit are also called gates . The size of the circuit isits number of gates. A Boolean threshold circuit is one whose gates are labelled by NOTs,unbounded degree ANDs, unbounded degree ORs, or unbounded degree thresholds TH n,k ,where TH n,k ( z , . . . , z n ) outputs 1 if, and only if, the number of 1’s in the input z , . . . , z n isat least k .If U is a G -set and W denotes the set of gates of C , we say that C is G -symmetric iffor every π ∈ G there exists σ ∈ Sym W such that C ( π,σ ) = C , where by C ( π,σ ) we meanthat the gates of the circuit are permuted according to σ , the labels from { x u } u ∈ U arepermuted according to π , and none of the other labels is moved. A circuit with U = L ( n )is called L -symmetric if it is Sym n -symmetric, with the natural action of Sym n on L ( n ).As for polytopes, we consider L -symmetric circuits as recognizing classes of L -structures onabstract sets V of vertices through bijections f : [ n ] → V . In the case of graphs, for example,in which L ( n ) = [ n ] , we say that such a circuit accepts a graph G with the set of vertices V of size n if for some, and hence every, bijection f : [ n ] → V it holds that C ( a ) = 1, where a is the vector that describes the image of G under f − , as in the previous section. In this section we prove the half of the equivalence that takes symmetric circuits with thresh-old gates into symmetric LPs. That is:
Lemma 1. If C is a class of L -structures that is recognized by a family of L -symmetricBoolean threshold circuits of size s ( n ) , then C is recognized by a family of L -symmetric LPlifts of size s ( n ) O (1) . In addition, if the Boolean circuits do not have threshold gates, thenthe size of the LP lifts is O ( s ( n )) . The main step in the construction is the simulation of the threshold gates. The na¨ıveapproach by which each threshold gate is replaced by an equivalent AND-OR-NOT circuitwill not work: it is known that any symmetric such circuit that computes the majorityfunction must have superpolynomial size. This follows from Theorem 2 in [3] and a standardEhrenfeucht-Fraiss´e argument. We need an alternative approach. As a step towards ourgoal, first we need to generalize the so-called Parity Polytope from Yannakakis [40].9 .1 The parity polytope explained
Yannakakis gives a polynomial-size symmetric polytope lift of the parity polytopePP( n ) := conv { ( x , . . . , x n ) ∈ { , } n : P nk =1 x k ≡ } . (1)Note that this polytope has the following interesting feature:if x , . . . , x n − are in { , } , then there exists a unique x n in R such that( x , . . . , x n ) is in PP( n ), and moreover this x n is the unique bit in { , } thatmakes the total sum P nk =1 x k odd. (2)For the existence just take x n ∈ { , } so that P nk =1 x k is odd. The uniqueness will followonce we show that any x n for which the extension vector ( x , . . . , x n ) belongs to PP( n ) is in { , } . In turn, this follows from the fact that all extreme points of PP( n ) are in { , } n , allhave the same parity, and a single bit-flip flips their parity. Indeed, if ( x , . . . , x n ) is in PP( n )but is not an extreme point, then it must be a non-trivial convex combination of at leasttwo extreme points and, whenever x , . . . , x n − are all in { , } , only two candidate extremepoints remain: ( x , . . . , x n − ,
0) and ( x , . . . , x n − , x i with 1 ≤ i ≤ n − n ).The construction of the lift of PP( n ) relies on the fact that the convex hulls of theHamming-weight slices of the n -dimensional Boolean hypercube are definable by a smalllinear program. Precisely, for each t ∈ [ n ] letEX( n, t ) := conv { ( x , . . . , x n ) ∈ { , } n : P nk =1 x k = t } . (3)Then it holds that the direct relaxation is tight:EX( n, t ) = { ( x , . . . , x n ) ∈ R n : P nk =1 x k = t, ≤ x k ≤ k = 1 , . . . , n } . (4)We provide an ± ǫ -proof since it is instructive. Let P be the polytope defined by the linearprogram in the right-hand side of (4). The inclusion EX( n, t ) ⊆ P is obvious. For theinclusion P ⊆ EX( n, t ) it suffices to show that every vertex of P has integral components;i.e., that every point in P that has some non-integral component fails to be a vertex of P because it is a non-trivial convex combination of two other points in P . Let x = ( x , . . . , x n )be a point in P , let I be the set of indices k ∈ [ n ] such that 0 < x k <
1, and assumethat I = ∅ . Now define ǫ := min { x k : k ∈ I } ∪ { − x k : k ∈ I } . (5)Thus ǫ >
0. Since P nk =1 x k = t and t is an integer, necessarily | I | ≥
2. Fix two differentindices i and j in I and, for b ∈ { , } , let x b = ( x b, , . . . , x b,n ) be the point defined by x b,k := x k + ( − b ǫ for k = ix b,k := x k − ( − b ǫ for k = jx b,k := x k for k ∈ [ n ] \ { i, j } .
10y the choice of ǫ we have 0 ≤ x b,k ≤ k ∈ [ n ] and every b ∈ { , } . Also n X k =1 x b,k = n X k =1 x k + ǫ − ǫ = n X k =1 x k = t (6)for both b ∈ { , } , and 12 ( x ,k + x ,k ) = 12 ( x k + x k ) = x k (7)for every k ∈ [ n ]. Since ǫ >
0, it follows that x and x are distinct points in P such that x = ( x + x ). Thus x is not a vertex of P .With the polytope EX( n, t ) in hand we are ready to describe the lift of PP( n ). First notethat a real vector x ∈ R n is in PP( n ) if, and only if, x = P t w t y t where each vector y t is inEX( n, t + 1), and the w t ’s are non-negative coefficients that add up to one, with t rangingover { , . . . , ⌊ n/ ⌋} . In order to express this as a linear program we introduce variables w t and z t,i for each t ∈ { , . . . , ⌊ n/ ⌋} and i ∈ { , . . . , n } with the intention that z t,i = w t y t,i for appropriate values y t,i that we do not care to actually get. Writing T for { , . . . , ⌊ n/ ⌋} and N for { , . . . , n } , the linear program that achieves this is the following: P t ∈ T w t = 10 ≤ w t ≤ t ∈ T P t ∈ T z t,i = x i for each i ∈ N P i ∈ N z t,i = (2 t + 1) w t for each t ∈ T ≤ z t,i ≤ w t for each t ∈ T and i ∈ N (8)The symmetry of this linear program with respect to the x -variables is obvious: given apermutation π ∈ Sym n , let σ be the permutation that maps z t,i to z t,π ( i ) and leaves each w t in place. For each integer n ≥
1, let | n | be the number of bits it takes to write n in binary. Wewant lifts of the following truncated parity polytopes defined for each pair of integers n ≥ q ∈ { , . . . , | n | − } :PP( n, q ) := conv { ( x , . . . , x n ) ∈ { , } n : ⌊ − q P nk =1 x k ⌋ ≡ } . (9)This time a vector x ∈ R n is in PP( n, q ) if, and only if, x = P t,r w t,r y t,r where eachvector y t,r is in EX( n, q (2 t + 1) + r ), and the w t,r are non-negative coefficients that add upto one, with ( t, r ) ranging over the set of pairs of integers with t ∈ { , . . . , ⌊ n/ q +1 ⌋} and r ∈ { , . . . , q − } . We introduce variables w t,r and z t,r,i for each ( t, r ) ∈ T and each i ∈ N ,where T = { , . . . , ⌊ n/ q +1 ⌋} × { , . . . , q − } and N = { , . . . , n } , with the intention that z t,r,i = w t,r y t,r,i for appropriate values y t,r,i that we do not care to actually get. The linear11rogram that achieves this is the following: P ( t,r ) ∈ T w t,r = 10 ≤ w t,r ≤ t, r ) ∈ T P ( t,r ) ∈ T z t,r,i = x i for each i ∈ N P i ∈ N z t,r,i = (2 q (2 t + 1) + r ) w t,r for each ( t, r ) ∈ T ≤ z t,r,i ≤ w t,r for each t ∈ T and i ∈ N (10)The symmetry of this linear program with respect to the x -variables is again obvious: given π ∈ Sym n , let σ map z t,r,i to z t,r,π ( i ) and leave every other variable in place. The poly-tope PP( n, q ) has the following interesting feature that is analogous to the one we arguedfor PP( n ):if x , . . . , x n − are in { , } and P n − k =1 x k ≡ − q ), then there exists aunique x n in R such that ( x , . . . , x n ) is in PP( n, q ), and moreover x n is theunique bit in { , } that makes the truncation ⌊ − q P nk =1 x k ⌋ odd. (11)For the existence just take x n ∈ { , } so that the truncation ⌊ − q P nk =1 x k ⌋ is odd, whichmust exist by the assumption that P n − k =1 x k ≡ − q ). The uniqueness follows once weshow that every x n for which the extension vector ( x , . . . , x n ) belongs to PP( n, q ) is in { , } ,and again the assumption that P n − k =1 x k ≡ − q ). For a proof that such an x n isin { , } it suffices to show that if ( x , . . . , x n ) ∈ PP( n, q ) satisfies the conditions, then it isan extreme point. If it were not an extreme point then it would be a non-trivial combinationof at least two extreme points and, whenever x , . . . , x n − are all in { , } , only two candidatesremain: ( x , . . . , x n − ,
0) and ( x , . . . , x n − , x i with 1 ≤ i ≤ n − P n − k =1 x k ≡ − q ), at least one ofthese extreme points must have even truncation ⌊ − q P nk =1 x k ⌋ , and hence not even belongto PP( n, q ); a contradiction. The goal in this subsection is to write a polynomial-size symmetric linear program that canbe used to simulate exact counting gates EX n,t ( x , . . . , x n ), which outputs 1 if the sum ofthe n input bits is exactly t , and 0 otherwise. In order to do this we use the truncated paritypolytopes to compute the bits of the binary representation of P nk =1 x i , and then comparethe result with the bits of the binary representation of t .First consider the following sequence of linear programs which depend only on n and12ot on t :( x , . . . , x n , , − z ) ∈ PP( n + 2 , x , . . . , x n , , , z , − z ) ∈ PP( n + 4 , x , . . . , x n , , , , , z , z , z , − z ) ∈ PP( n + 8 , x , . . . , x n , (2 q ) , z (1)1 , z (2)2 , . . . , z (2 q − ) q , − z q +1 ) ∈ PP( n + 2 q +1 , q )...( x , . . . , x n , (2 | n |− ) , z (1)1 , z (2)2 , . . . , z (2 | n |− ) | n |− , − z | n | ) ∈ PP( n + 2 | n | , | n | − , (12)where, for ℓ ≥
1, the notation a ( ℓ ) denotes the string a, a, . . . , a of length ℓ . We claim thefollowing property:if x , . . . , x n are in { , } , then there is a unique vector ( z , . . . , z | n | ) ∈ R | n | which together with x , . . . , x n is a solution, and in this solution z k ∈ { , } for all k , and P nk =1 x k = P | n | k =1 (1 − z k )2 k − ; in other words, z , . . . , z | n | are theflips of the bits of the binary representation of P nk =1 x k , listed from least tomost significant bit. (13)From now on in this proof, let X = P nk =1 x k . The first part of the claim follows from thecorresponding property of PP( n, q )’s, and induction on q = 0 , , . . . , | n | −
1. The second partis proved by showing that z , . . . , z q +1 are all in { , } and X ≡ q +1 X k =1 (1 − z k )2 k − (mod 2 q +1 ) (14)also by induction on q = 0 , , . . . , | n | −
1. For q = 0 the claim is true since, by the mainproperty of PP( n + 2 ,
0) = PP( n + 2), there is a unique z for which ( x , . . . , x n , , − z ) isin PP( n + 2), and this is the unique bit that makes X + 1 + 1 − z odd, hence X − z odd.Assume now that z , . . . , z q +1 are all in { , } and that (14) holds for q ∈ { , . . . , | n | − } and we prove it for q + 1. First observe that X + 2 q +1 + q +1 X k =1 z k k − ≡ X + q +1 X k =1 z k k − ≡ q +1 − ≡ − q +1 ) , (15)where in the second equivalence we used the induction hypothesis on q . Moreover z , . . . , z q +1 are all in { , } , also by induction hypothesis on q . Thus, the vector( x , . . . , x n , (2 q +1 ) , z (1)1 , z (2)2 , . . . , z (2 q +1 − ) q +1 , − z q +2 ) (16)satisfies the hypothesis of property (13) for PP( n + 2 q +2 , q + 1). It follows that there is aunique z q +2 that puts (16) in PP( n + 2 q +2 , q + 1), and this is the unique bit such that thequantity ⌊ − ( q +1) ( X + 2 q +1 + q +1 X k =1 z k k − + 1 − z q +2 ) ⌋ (17)13s odd. Now, (14) says that X = 2 q +1 ⌊ − ( q +1) X ⌋ + q +1 X k =1 (1 − z k )2 k − , (18)which means that the unique bit that makes the quantity in (17) odd is the flip of the( q + 1)-th least significant bit of X .Now, exact- t counting gates can be expressed using an additional linear program thatsimulates an AND gate to compare the bits z , . . . , z | n | with (the flips of) the bits of thebinary representation of t . For both b ∈ { , } , let K b ⊆ [ | n | ] be the subset of bit-positionsat which the | n | -long binary representation of t is b . Then, the relation y = EX n,t ( x , . . . , x n )is expressed by the union of (12) and the following: y ≥ P k ∈ K z k + P k ∈ K (1 − z k ) − | n | + 1 y ≤ z k for every k ∈ K y ≤ − z k for every k ∈ K ≤ y ≤ . (19)We write EX n,t ( x , . . . , x n , y ) to denote the LP that has all the constraints in (12) and allthe constraints in (19). We summarize its main properties in the following: Lemma 2.
The linear program EX n,t ( x , . . . , x n , y ) has size polynomial in n , is symmetricwith respect to the group of permutations of x , . . . , x n , y that fix y , and has the followingproperty: If x , . . . , x n are all in { , } , then there is a unique y ∈ R such that ( x , . . . , x n , y ) can be extended to a feasible solution, and this y is the unique output bit of the correspondinggate evaluated on inputs x , . . . , x n .Proof. The bound on the size follows by inspection. The symmetry with respect to thegroup of permutations of x , . . . , x n , y that fix y follows from the symmetry claims for thetruncated parity polytopes, together with the extension that keeps each z i -variable in place.For proving the main property, assume that x , . . . , x n are all in { , } . The first part (12)of EX n,t ( x , . . . , x n , y ) has the feature expressed in property (13). Let then z , . . . , z | n | sat-isfy the conclusion in that property. Thus, z , . . . , z | n | are the flips of the bits of the bi-nary representation of P ni =1 x i . In particular they are all in { , } . The second part (19)of EX n,t ( x , . . . , x n , y ) has the feature that if all z , . . . , z | n | are in { , } , then there is aunique y in R that makes a solution, and this y is precisely the bit that is 1 if and only ifall z k with k ∈ K are one, and all z k with k ∈ K are zero. In other words, y = 1 if andonly if z , . . . , z | n | are the flips of the bits of the binary representation of t , and hence if andonly if P nk =1 x k = t . 14 .4 The construction Let us recall how AND, OR and NOT gates are represented by LPs. The LPs for these threetypes of gates do not require auxiliary variables, and their size is linear in the fan-in. Define:AND( x , . . . , x n , y ) OR( x , . . . , x n , y ) NOT( x, y ) y ≥ P ni =1 x i − n + 1 1 − y ≥ P ni =1 (1 − x i ) − n + 1 y = 1 − xy ≤ x i − y ≤ − x i ≤ x ≤ ≤ x i ≤ ≤ x i ≤ ≤ y ≤ . ≤ y ≤ . ≤ y ≤ . The main properties of these LPs are summarized in the following:
Lemma 3.
The linear programs
AND( x , . . . , x n , y ) , OR( x , . . . , x n , y ) , and NOT( x , y ) havesize linear in n , are symmetric with respect to the group of permutations of its variables thatfix y , and have the following property: If x , . . . , x n are all in { , } , with n = 1 for NOT ,then there is a unique y ∈ R that makes ( x , . . . , x n , y ) feasible, and this y is the uniqueoutput bit of the corresponding gate evaluated on inputs x , . . . , x n .Proof. For NOT this is totally obvious. For AND it is easy to check, and for OR it followsfrom the corresponding properties of AND and NOT.We define the conversion from circuits to LPs. Let U be a set and let C be a circuitwith inputs ( x u ) u ∈ U . Let C ′ be the circuit that results from replacing each k -threshold gatewith inputs y , . . . , y m by W mt = k EX m,t ( y , . . . , y m ), where EX m,t ( y , . . . , y m ) denotes an exactcounting gate with inputs y , . . . , y m which outputs 1 if and only if the exact number of 1’s inthe input is t . The resulting circuit is equivalent to C and its number of gates is polynomialin that of C . We define the LP; we call it LP( C ).For each gate i in C ′ , let y i be a variable constrained by the inequalities0 ≤ y i ≤ . (20)For each gate o in C ′ add the constraints and the auxiliary variables, when necessary, thatexpress their computation: y o = x u if o is an input gate labelled by x u , NOT( y i , y o ) if o is a NOT gate with input i, AND( y i , . . . , y i m , y o ) if o is an AND gate with inputs i , . . . , i m , OR( y i , . . . , y i m , y o ) if o is an OR gate with inputs i , . . . , i m , EX m,t ( y i , . . . , y i m , y o ) if o is an EX m,t gate with inputs i , . . . , i m ,y o = 1 if o is the output gate of the circuit . (21)By Lemmas 2 and 3, all six cases have size polynomial in the number of inputs, hence thetotal size is polynomial in the size of C ′ . In case C does not have TH gates, the step forreplacing them by EX gates is not done, and all gates are AND, OR, NOT, so the size ofthe LP stays linear in the size of C . 15 emma 4. If U is a G -set and C is G -symmetric, then LP( C ) is G -symmetric and recognizesthe same subset of { , } U as C .Proof. The claim that LP( C ) recognizes the same subset of { , } U as C follows from Lem-mas 2 and 3. We prove the symmetry. First note that the intermediate circuit C ′ is also G -symmetric. Now we show that the LP is also G -symmetric. Fix some π ∈ G . Let σ be apermutation of the gates of C ′ so that the pair ( π, σ ) leaves C ′ in place. In particular, foreach gate o of C ′ with inputs i , . . . , i m , if p = σ ( o ), then p is the same type of gate as o , hasthe same fan-in m , and if o is an input gate fed by x u , then p is an input gate fed by x π ( u ) .Moreover, if j , . . . , j m are the inputs of gate p , then there is a permutation τ o ∈ Sym m sothat σ ( i k ) = j τ o ( k ) for every k ∈ [ m ]. If we think of σ as mapping the output variable y o ofthe linear program P o = P ( y i , . . . , y i m , y o ) for gate o to the output variable y p of the linearprogram P p = P ( y j , . . . , y j m , y p ) for gate p , then, by the above, this map takes the inputvariables of P o to the input variables of P p . We want to show that this σ can be extendedto also map the auxiliary variables of P o to the auxiliary variables of P p in such a way thatthe resulting extension of π is an automorphism of the linear program P for C ′ . We definethis extension automorphism gate by gate.We start with the internal gates of C ′ . By the symmetry claims in Lemmas 2 and 3,the permutation that agrees with τ o on the input variables y j , . . . , y j m of P p and that fixesthe output variable y p , extends to an automorphism ρ o of P p . With the automorphisms ρ o in hand, we are ready to define the automorphism of P that extends π : for each gate o ,map the variable y o to y σ ( o ) , and map each auxiliary variable of P o to the image under ρ o ofthe corresponding auxiliary variable of P p , where p = σ ( o ). For all internal gates, this hasthe required properties by construction. For the gates o that are fed by a variable x u , thegate σ ( o ) must be fed by the variable x π ( u ) , and therefore y o = x u gets mapped to y σ ( o ) = x π ( u ) , as required. For the output gate o of C ′ we have σ ( o ) = o , and the constraint y o = 1gets mapped to itself. Proof of Lemma 1.
This is a consequence of Lemma 4: for the n -th LP we let U = L ( n ), let G = Sym n with the natural action on L ( n ), and use LP( C n ), where C n is the n -th circuit. We say that a function s ( n ) is at most weakly exponential if there exists a positive real ǫ such that s ( n ) ≤ n − ǫ for every sufficiently large n . In this section we prove the second halfof the equivalence which takes families of symmetric linear programs to families of formulasof counting logic. That is: Lemma 5. If C is a class of L -structures that is recognized by a family of L -symmetricpolytope lifts of size s ( n ) , then C is recognized by a family of C k ( n ) formulas, where k ( n ) = O (log( s ( n )) / (log( n ) − log log( s ( n )))) . Moreover, if s ( n ) is at most weakly exponential, thenthe formulas have size s ( n ) O (1) . bounded support . The existence of boundedsupports implies that a property of L -structures recognised by a family of L -symmetric LPlifts is recognised by a family of manageable such LP lifts. Let us illustrate the notion of amanageable linear program by an example.Consider a graph-symmetric polytope P over R [2] × R given by the following system oflinear constraints: x ≤ x ≤ x − y ≤ x − y ≤ y + y ≤ P are central to the notion of a manageable polytope. Firstly, each ofthe auxiliary variables y , y is essentially a tuple of integers from [2] of a bounded length.Secondly, P is graph-symmetric with respect to the natural action of the group Sym on theset { y , y } . Indeed, for any permutation π ∈ Sym the system of constraints: x π (1) π (1) ≤ x π (2) π (2) ≤ x π (1) π (2) − y π (1) π (2) ≤ x π (2) π (1) − y π (2) π (1) ≤ y π (1) π (2) + y π (2) π (1) ≤ L -structure over an n -element domain, a manageable polytope lift over R L ( n ) × R W can be turned into a linear program whose variables are indexed by tuples of elementsof the structure of bounded length. Moreover, the symmetry condition guarantees that thiscan be done without referring to any concrete bijection between [ n ] and the domain of thestructure. Thus, as we show, it can be performed by means of logical interpretations.In what follows, from a family ( P n ) n ∈ N of L -symmetric LP lifts we obtain a family ( ¯ P n ) n ∈ N of manageable LP lifts recognising the same family of structures. Lemma 6 below impliesthat from each P n one can construct a polytope lift b P n which recognises the same subset of { , } L ( n ) but comes with an action of the group Sym n witnessing its symmetry. Further,in Subsection 4.2 we show that the action of Sym n on each of the constraints and auxiliaryvariables of b P n depends on a subset of [ n ] of bounded size called its support. In the secondpart of Subsection 4.2 we analyse properties of sets whose elements have bounded supportsin order to show that they are essentially sets of tuples of integers from [ n ]. This implies, inSubsection 4.3, that each b P n after a small modification becomes a manageable LP lift ¯ P n .Finally, in Subsection 4.4 based on ¯ P n we construct a FOC-interpretation that given an L -structure A over an n -element domain outputs a linear program which has a solution if and17nly if A belongs to the class of interest (for a definition of an interpretation see e.g. [24]).Since solving linear programs is expressible in FPC [4], we are able to conclude the proof. In this subsection we consider general G -symmetric LPs, i.e., not necessarily L -symmetric.Let U be a G -set and let P ⊆ R U × R W be a G -symmetric polytope given by a sequenceof linear constraints ( γ v ) v ∈ V where each γ v is of the form a T x + b T y ≤ c , with x = ( x u ) u ∈ U and y = ( y w ) w ∈ W . We say that the polytope P is rigid if for every π ∈ G there exists aunique element of Sym W , let us denote it by σ π , such that P ( π,σ π ) = P .Assume that P is rigid. It is easy see that the mapping from G to Sym W given by π σ π is a group homomorphism. Hence, there is a natural action of the group G on the set ofauxiliary variables { y w } w ∈ W such that for any π ∈ G and w ∈ W applying π to y w gives y σ π ( w ) , and the polytope P is G -symmetric with respect to this action. Moreover, this inducesan action of the group G on the set of linear constraints { γ v } v ∈ V in the obvious way: for any π ∈ G and any v ∈ V applying π to γ v of the form a T x + b T y ≤ c gives a T x π + b T y σ π ≤ c ,and the symmetry of P guarantees that this is also a constraint. For rigid G -symmetricpolytopes, we write y π to mean y σ π , we use γ πv to denote a T x π + b T y σ π ≤ c , and P π todenote P ( π,σ π ) .Suppose that a subset A of { , } U is recognised by a G -symmetric polytope lift P .We show that there exists a rigid G -symmetric polytope lift b P of size polynomial in thesize of P recognising A . More precisely, the number of auxiliary variables and the numberof constraints of b P are at most, respectively, the number of auxiliary variables and thenumber of constraints of P , and the bit-size of the coefficients which appear in the linearconstraints defining b P is polynomial in the bit-size of the coefficients which appear in thelinear constraints defining P .The construction of b P goes as follows. For the subgroup of Sym W consisting of all permu-tations σ such that P (id ,σ ) = P , consider the orbits of the set of auxiliary variables { y w } w ∈ W under the action of this subgroup. By identifying the variables in each of those orbits weobtain a new G -symmetric polytope lift recognising A with potentially smaller number ofauxiliary variables. This procedure needs to be iterated until the obtained polytope is rigid.Let us provide more details.We need a few pieces of notation. For every π ∈ G , by ext( π ) we denote the set ofall σ ∈ Sym W satisfying P ( π,σ ) = P and by H we denote the union of ext( π ) over all theelements π of G , i.e., H = { σ ∈ Sym W : there exists π ∈ G such that P ( π,σ ) = P } . Itis easy to see that both ext(id) and H are subgroups of Sym W . Moreover, for any π ∈ G the set ext( π ) is a coset of ext(id) in H . Now let K = P π ∈ G { π } × ext( π ). Observe that K is a subgroup of the direct product G × H and consider the projection homomorphism f G : K → G given by f G ( π, σ ) = π . The kernel of this homomorphism is the group { id } × ext(id). Let us denote it by J . Since the homomorphism f G is surjective, the quotient K/J is isomorphic to G .Notice that the polytope P is rigid if, and only if, the group ext(id) is trivial. Assumetherefore that this is not the case and let O denote the set of orbits of the set of auxiliary18ariables { y w } w ∈ W under the action of ext(id). Recall that O forms a partition of { y w } w ∈ W .For every orbit O ∈ O , we introduce a new variable y O and for any polytope R , let b R denote the polytope obtained from R by substituting, for each O ∈ O , every variable in O (if present) by y O . We aim to show that b P is a G -symmetric polytope recognizing A .The projection homomorphism f H : K → H given by f H ( π, σ ) = σ can be seen as ahomomorphism from K to Sym W and hence defines an action of the group K on the set ofvariables { y w } w ∈ W . Since J is a normal subgroup of K , the quotient group K/J acts on theset of orbits of { y w } w ∈ W under the (induced) action of J : for any π ∈ G , σ ∈ ext( π ) and w ∈ W , applying ( π, σ ) J to the orbit of y w maps it to the orbit of y σ ( w ) . Since the groups J and ext(id) as well as the groups K/J and G are isomorphic, this gives us an action of G on O which, as we argue below, witnesses the fact that the polytope b P is G -symmetric.Unfolding the abstract definition of this group action, there is a homomorphism h from thegroup G to the symmetric group Sym O such that for every π ∈ G , h ( π ) is the permutation of O , which for any w ∈ W , maps the orbit of the variable y w to the orbit of the variable y σ ( w ) ,where σ is any permutation from ext( π ). Finally, let us observe that, for any π ∈ G and σ ∈ ext( π ), it holds that b P ( π,h ( π )) = \ P ( π,σ ) = b P , which means that indeed b P is G -symmetric.It remains to show that b P recognises A .Let x ∈ A and take some y ∈ R W such that ( x , y ) ∈ P . Note that for every σ ∈ ext(id) itholds that ( x , y σ ) ∈ P . Hence, we have ( x , y ′ ) ∈ P where y ′ = P σ ∈ ext(id) y σ / | ext(id) | . Nowfor every O ∈ O let p O be the sum of the values taken by the variables from O in the solution( x , y ). In the solution ( x , y ′ ) every variable y in O takes value p O | ext(id) y | / | ext(id) | = p O / |O| , where ext(id) y denotes the stabiliser of y in ext(id). Hence, by assigning for every O ∈ O the value p O to the variable y O we obtain a point ( x , b y ) in b P . This implies that A iscontained in the subset of { , } U recognised by b P . The other inclusion is clear.We are now ready to state the main conclusion of this subsection. Lemma 6.
For every G -symmetric polytope P of size s , there is a rigid G -symmetric polytope Q which recognises the same set and the size of Q is not more than s log( s ) .Proof. First note that for any G -symmetric polytope R , if R is not rigid, then b R has strictlyfewer variables than R . Thus, starting at P , if we iterate the process, we must, in a finitenumber of steps reach a rigid polytope Q . For the size bound, note that the number ofvariables and constraints in Q is at most the corresponding number in P . Moreover, eachcoefficient in Q is the sum of distinct coefficients in P , of which there are at most s . Thus,if each coefficient in P can be expressed with b bits, each coefficient in Q requires at most b log( s ) bits and the bound follows. For a Sym n -set Y , a subset S of [ n ] is said to be a support of an element y ∈ Y if for every π ∈ Sym n that fixes S pointwise, it holds that π · y = y . Intuitively, this means that theaction of the group Sym n on y depends only on the set S . We get even supports by replacingthe symmetric group Sym n by the alternating group Alt n . A subset S of [ n ] is said to be an19 ven support of y ∈ Y if for every π ∈ Alt n that fixes S pointwise, we have π · y = y . Clearly,any support of y is also an even support of y . Note also that every element of a Sym n -set issupported by the whole set [ n ].For a non-negative integer k , an (even) support S is called k - bounded if | S | ≤ k . A Sym n -set Y is called k - supported if each element of Y has a k -bounded support. An L -symmetricpolytope P is called k - supported if the set of auxiliary variables and the set of constraints of P are k -supported. We now show the following: Lemma 7.
There exists a positive integer n such that for any positive integers s and n satisfying s ≥ n ≥ n , the following holds: If P is a rigid L -symmetric LP lift of size s forstructures with n elements, then P is k -supported, where k = O (log( s ) / (log( n ) − log log( s ))) .Moreover, if s ≤ n/ , then the size of P is at most n k .Proof. For the sake of simplicity we give the proof for the case when L consists of a singlebinary symbol, that is for the case of graphs. The general case is completely analogous.Consider a rigid graph-symmetric polytope lift P ⊆ R [ n ] × R W of size s which recognisessome property of graphs with n vertices, and let P be given by a sequence of linear constraints( γ v ) v ∈ V where each γ v is of the form a T x + b T y ≤ c , with x = ( x ij ) i,j ∈ [ n ] and y = ( y w ) w ∈ W .Since P is rigid, it comes with an action of the group Sym n on the set of auxiliary variables { y w } w ∈ W such that for every π ∈ Sym n we have P π = P , and with an induced action of thegroup Sym n on the set of constraints { γ v } v ∈ V .If the size s of the linear program P is greater than 2 n/ , we can take k = n . Observethat in case s > n/ we havelog( s ( n ))log( n ) − log log( s ( n )) ≥ n/d log( n ) − log( n ) + log(3) = n . (22)It follows that k = O (log( s ) / (log( n ) − log log( s ))). Since any element of a Sym n -set issupported by [ n ], for this choice of k , every auxiliary variable and every constraint of P hasa k -bounded support.In the case s ≤ n/ the argument is more involved. First we obtain bounded evensupports. Take t = log( s ) / (log( n ) − log log( s )) and k = ⌈ t ⌉ . Observe that the denominatorin the definition of t is non-zero, since by assumption we have s ≤ n/ < n . Also, we have0 < t ≤ k ≤ n/ < n/ < n/e , to be used later in the proof. Let us start by notingthat t log (cid:16) nt (cid:17) = log( s ) log( n ) − log log( s ) + log(log( n ) − log log( s ))log( n ) − log log( s ) > log( s ) . (23)The inequality follows from the fact that the big fraction in the middle is strictly biggerthan 1 since s ≤ n/ .For any S ⊆ [ n ], let Alt ( S ) denote the group of all even permutations of [ n ] that fix theset S pointwise. We use the following fact which guarantees that subgroups of the symmetricgroup with small index contain as subgroups large alternating groups. Lemma 8 (see Theorem 5.2B in [16]) . If n > and ≤ k ≤ n/ , and G is a subgroup of Sym n with [Sym n : G ] < (cid:0) nk (cid:1) , then there is a set S ⊆ [ n ] with | S | < k such that Alt ( S ) ≤ G . w ∈ W , let St w denote the stabilizer of y w in Sym n , i.e., the subgroup of Sym n definedby St w = { π ∈ Sym n : π · y w = y w } . Since [Sym n : St w ] is the size of the orbit of y w underthe action of Sym n and the total number of auxiliary variables is bounded by the size of P ,we have [Sym n : St i ] ≤ s < (cid:16) nt (cid:17) t ≤ (cid:16) nk (cid:17) k ≤ (cid:18) nk (cid:19) (24)with the second following from (23), and the third from 0 < t ≤ k < n/ < n/e and thefact that f ( x ) = ( n/x ) x is an increasing function of x in the interval (0 , n/e ). Lemma 8implies that, if n is large enough, there exist S ⊆ [ n ] with | S | < k and Alt ( S ) ≤ St w . Thisis a k -bounded even support of y w in the way we defined. An entirely analogous argumentyields a k -bounded even support for each constraint in { γ v } v ∈ V .In order to obtain supports in place of even supports we need to introduce a way oflooking at automorphism groups of symmetric polytopes as automorphism groups of graphs.In the following, whenever we talk about coloured vertices, by a colour we mean a graphgadget that forces the vertices of a colour to be mapped to vertices of the same colour byevery possible automorphism.Let C be the set of all different numerical coefficients which appear in the linear con-straints representing P . A graph representation of the polytope P is a graph P with:1. five disjoint sets of vertices: [ n ], { x ij } i,j ∈ [ n ] , { y w } w ∈ W , { γ v } v ∈ V and C ,2. the vertices in [ n ], { x ij } i,j ∈ [ n ] , { y w } w ∈ W and { γ v } v ∈ V coloured with four different coloursdepending on the set they belong to,3. each vertex in C coloured with a unique colour, which does not appear anywhere elsein the graph,4. for any i, j ∈ [ n ], an edge from i to x ij and from j to x ij ,5. for any v ∈ V and any variable which appears in the constraint γ v , a pair of edges:from the variable to its coefficient, and from this coefficient to γ v ,6. for any v ∈ V , an edge from γ v to the vertex in C corresponding to its constant term.Observe that the automorphism group of the graph P is isomorphic to the automorphismgroup of P . Also, the number of vertices in P can be bounded by O ( s ). This is becausethe number of vertices introduced in item (1) is at most s and each gadget introduced in (2)and (3) can be chosen to have at most 2 s vertices and there are at most s of them.For any S ⊆ [ n ], let Sym ( S ) denote the group of all permutations of [ n ] that fix theset S pointwise. Take some w ∈ W and let S be a k -bounded even support of y w . SinceAlt ( S ) ≤ St w , we have Alt ( S ) ≤ St w ∩ Sym ( S ) ≤ Sym ( S ) . Hence, St w ∩ Sym ( S ) = Alt ( S ) or St w ∩ Sym ( S ) = Sym ( S ) . We argue it is the latter case that holds using the followingtheorem which states that a graph whose automorphism group is the alternating group onan n -element set must be of size exponential in n . Lemma 9 (Theorem A in [30]) . If n > , then the number of vertices of any graph whosefull automorphism group is isomorphic to Alt n is at least / (cid:0) n ⌊ n/ ⌋ (cid:1) ∼ n / √ πn . w ∩ Sym ( S ) = Alt ( S ) . We use Lemma 9 to arrive at a contradiction.Consider the graph representation P of the polytope P modified in the following way: thevertices in S ⊆ [ n ] and the vertex y w are coloured each with a different colour which did notappear in the graph before. Observe that the automorphism group of the graph P w obtainedthis way is isomorphic to St w ∩ Sym ( S ) , and therefore isomorphic to Alt ( S ) , which in turn isisomorphic to the alternating group on the set [ n − | S | ]. And, once again, the number ofvertices of P w is O ( s ). Thus, if n is large enough, we have s < n/ < (cid:18) n ⌊ n/ ⌋ (cid:19) . (25)Hence, by Lemma 9, we obtain the desired contradiction.From St w ∩ Sym ( S ) = Sym ( S ) it follows that Sym ( S ) ≤ St w , which means that S is a k -bounded support of y w in the way we defined. An analogous argument yields a k -boundedsupport for each linear constraint in { γ v } v ∈ V . Note also that that s < (cid:0) nk (cid:1) ≤ n k . In particular, P has at most n k auxiliary variables, at most n k constraints, and all its coefficients andconstant terms can be written down using at most n k bits.We now show that it is possible to (non-uniquely) represent the auxiliary variables andconstraints of k -supported polytopes by tuples of integers from [ n ] of length k in a way thatis consistent with the group action. In order for the representation to be uniform across all n , we extend the definition of the set [ n ] ( k ) to the case when k > n . For positive integers n and k such that k > n , the set [ n ] ( k ) consists of k -tuples of elements of [ n ] with the first n components pairwise distinct and the last k − n components equal to the n -th component.In particular, if k > n , then every tuple in [ n ] ( k ) contains all the elements of [ n ].For any positive integer n and any non-negative integer k , we consider the set [ n ] ( k ) as aSym n -set with the natural action of the group Sym n . Note that this Sym n -set has one orbit.Indeed, if we take any k -tuples s and s from [ n ] ( k ) , then there exists a permutation π suchthat π · s = s . This is because the equality types of any two elements of [ n ] ( k ) are the same. Lemma 10.
Let Y be a single-orbit k -supported Sym n -set. There is a surjective homomor-phism from [ n ] ( k ) to Y .Proof. Take any y ∈ Y and let S be a k -bounded support of y . Since a superset of a supportis a support itself, without loss of generality we can assume that | S | = min { k, n } . Now, picka tuple s ∈ S ( k ) . If k ≤ n , then | S | = k and every element of S appears exactly once in thetuple s , otherwise the tuple s contains all the elements of [ n ].We define a homomorphism f from the Sym n -set [ n ] ( k ) to the Sym n -set Y which for any π ∈ Sym n maps π · s to π · y . The only thing that needs to be verified is whether the function f is well defined. To this end, suppose that for some permutations π , π ∈ Sym n it holdsthat π · s = π · s . Then π − π · s = s , that is, the permutation π − π fixes the support S of y pointwise. Therefore, π − π · y = y which implies that π · y = π · y . Since Y has oneorbit, the homomorphism f is surjective. 22nce a surjective homomorphism f from a Sym n -set [ n ] ( k ) to a Sym n -set Y is fixed, thefamily { f − ( y ) } y ∈ Y forms a partition of [ n ] ( k ) . Hence, for any y ∈ Y , each tuple ( i , . . . , i k )from f − ( y ) uniquely identifies y , and is called an identifier of y (with respect to the homo-morphism f ). In most cases each element of Y has several identifiers.Let us illustrate Lemma 10 by an example. For n ≥
2, consider the set Y of two-element subsets of [ n ] with the natural action of the group Sym n , i.e., for any π ∈ Sym n and any distinct i, j ∈ [ n ], we have π · { i, j } = { π ( i ) , π ( j ) } . This single-orbit Sym n -set is k -supported, for any k ≥
2. Let us take k = 2. The homomorphism from Lemma 10 is thenunique and given by ( i, j )
7→ { i, j } . Note that the inverse image of any { i, j } ∈ Y has twoelements, and hence, each element { i, j } of Y has two identifiers: ( i, j ) and ( j, i ). For k = 3,applying Lemma 10 yields several different homomorphisms. If n ≥
3, one of them is givenby ( i, j, ∗ )
7→ { i, j } , where by ∗ we mean any element of [ n ] distinct from both i and j . Inthis case, the inverse image of any { i, j } ∈ Y has 2 n − n = 2 and k = 3,the homomorphism is again unique and given by ( i, j, j )
7→ { i, j } yielding two identifiers, forevery element of Y .To give one more example, consider a single-orbit set which consists of a single element y with the trivial action of the symmetric group Sym n . This set is k -supported, for anynon-negative integer k . For any k ≥
0, the homomorphism from Lemma 10 mapa each tuplefrom [ n ] ( k ) to y . In particular, for k = 0, we have ǫ y , where ǫ is the only element of [ n ] (0) ,that is, the empty tuple.To represent elements of a k -supported Sym n -sets with potentially more than one orbit,we need to introduce several copies of the set [ n ] ( k ) , one for each orbit. Corollary 1.
Let Y be a k -supported Sym n -set. There is a surjective homomorphism from Q × [ n ] k to Y , where the size of Q is equal to the number of orbits of Y . It is clear how to extend the definition of an identifier to the general case discussed in thecorollary above. Note that if a tuple ( q, i , . . . , i k ) is an identifier of y ∈ Y , then the tuple( q, π ( i ) , . . . , π ( i k )) is an identifier of π · y . For a non-negative integer k , a polytope P over R L ( n ) × R W is called k - manageable if:1. there are two sets Q and T with a trivial action of the group Sym n ,2. the set of constraints of P is indexed by V = Q × [ n ] k ,3. the set of auxiliary variables of P is indexed by W = T × [ n ] k ,4. P is L -symmetric with respect to the natural action of Sym n on W , and the inducedaction of Sym n on the set of constraints is exactly the natural action of Sym n on V .An example of a manageable polytope is given at the very beginning of this section. Therethe set of auxiliary variables has only one orbit { y , y } so introducing T is not necessary.The key property of k -manageable polytopes, which allows us to use them in the trans-lation from families of linear programs to logic, is the following.23 emma 11. If P is a k -manageable polytope with constraints indexed by V = Q × [ n ] ( k ) and auxiliary variables indexed by W = T × [ n ] ( k ) , then for any R ∈ L , q ∈ Q , t ∈ T , i , i ′ , j , j ′ ∈ [ n ] ( k ) , k , k ′ ∈ [ n ] ar( R ) :1. the constant terms of the linear constraints γ ( q, i ) and γ ( q, i ′ ) are the same,2. if the equality types of the tuples ( j , i ) and ( j ′ , i ′ ) are the same, then the coefficientof the variable y ( t, j ) in the linear constraint γ ( q, i ) is the same as the coefficient of thevariable y ( t, j ′ ) in the linear constraint γ ( q, i ′ ) ,3. if the equality types of the tuples ( k , i ) and ( k ′ , i ′ ) are the same, then the coefficientof the variable x ( R, k ) in the linear constraint γ ( q, i ) is the same as the coefficient of thevariable x ( R, k ′ ) in the linear constraint γ ( q, i ′ ) .Proof.
1. Recall that every tuple in the set [ n ] ( k ) has the same equality type. Therefore,there exists a permutation π ∈ Sym n which maps the tuple i to i ′ . Since γ π ( q, i ) = γ ( q, i ′ ) , theconstant terms in the linear constraints γ ( q, i ) and γ ( q, i ′ ) are the same.2. If the equality types of the tuples ( j , i ) and ( j ′ , i ′ ) are the same, then there exists apermutation π ∈ Sym n , which maps j to j ′ , and i to i ′ . Let a be the coefficient of the variable y ( t, j ) in the linear constraint γ ( q, i ) . By applying the permutation π to the constraint γ ( q, i ) , weget the constraint γ ( q, i ′ ) . Moreover, since π · y ( t, j ) = y ( t, j ′ ) , the coefficient of the variable y ( t, j ′ ) in γ ( q, i ′ ) is a . The proof of 3. is analogous.Now, suppose that a k -supported rigid L -symmetric LP lift P ⊆ R L ( n ) × R W recognizessome property of L -structures, that is, a subset A of { , } L ( n ) . We argue that there existsa k -manageable polytope lift ¯ P recognising A . Since the polytope P is k -supported, byapplying Lemma 10 we obtain two sets of identifiers: ¯ V = Q × [ n ] k for the constraints, and¯ W = T × [ n ] k for the auxiliary variables. Let us introduce a new variable of the form y ( t, j ) ,for any identifier ( t, j ) ∈ ¯ W . We obtain a manageable polytope ¯ P from the polytope P by first, replacing, for each w ∈ W , the auxiliary variable y w in P by the sum of variables y ( t, j ) over the set of all identifiers ( t, j ) of y w ; and secondly, replacing, for every v ∈ V , theconstraint γ v , by several copies of this constraint, one for every identifier ( q, i ) of γ v . Theobtained polytope lift ¯ P is clearly k -manageable and it is easy to see that it recognizes thesame property of L -structures.To summarize, the constructions in Subsections 4.1, 4.2 and 4.3 imply that if a prop-erty of L -structures is recognised by a family of L -symmetric LP lifts of size s ( n ), thenthe same property is recognised by a family of k ( n )-manageable LP lifts, where k ( n ) = O (log( s ( n )) / (log( n ) − log log( s ( n )))). We now put everything together in the proof of Lemma 5. For the sake of simplicity wegive the proof for the case when L consists of a single binary symbol, that is for the case ofgraphs. The general case is completely analogous.24et P ⊆ R [ n ] × R W be a graph-symmetric LP lift of size s recognising some property ofgraphs with n vertices, that is, a subset A of { , } [ n ] . We show that the same property ofgraphs is definable by a C k formula, where k = O (log( s ) / (log( n ) − log log( s ))).Let b P be a rigid graph-symmetric LP lift recognising A , as constructed in Subsection 4.1.Recall that its size s ′ is at most s log( s ) where s is the size of P . In particular, s ′ ≤ s .If s > n/ , by a calculation analogous to (22) at the beginning of Subsection 4.2, we have n = O (log( s ) / (log( n ) − log log( s ))). Since every class of graphs with n vertices is definablein C n , we complete the proof of the lemma in this case by taking k = n .If s ≤ n/ , then s ′ ≤ s ≤ n/ . Hence, by Lemma 7, for some k = O (log( s ′ ) / (log( n ) − log log( s ′ ))), b P is k -supported, has at most n k auxiliary variables, at most n k constraints,and all its coefficients and constant terms can be encoded using at most n k bits. Moreover,any such k clearly satisfies k = O (log( s ) / (log( n ) − log log( s ))).Let ¯ P be a k -manageable polytope lift recognising A (as described in Subsection 4.3)with the set of constraints indexed by Q × [ n ] k , and the set of auxiliary variables indexed by T × [ n ] k . Note that it follows from the construction of ¯ P that the number of elements in thesets T and Q is bounded, respectively, by the number of auxiliary variables and the numberof constraints in b P . Hence, | Q | , | T | ≤ n k .Suppose now that we are given a graph G with the set of vertices V of size n and the setof edges E . Intuitively, if we could fix a bijection between [ n ] and V , we could then computefrom ¯ P and G a linear program ¯ P G with the set of constraints I and the set of variables J as follows: I = { γ ( q, v ) : q ∈ Q, v ∈ V k } ,J = { x vw : v, w ∈ V } ∪ { y ( t, v ) : t ∈ T, v ∈ V k } . In order to decide whether G has the property of interest we would then check if thepartial valuation: x vw = 1 if ( v, w ) ∈ E , and x vw = 0 otherwise, can be extended to afull solution. This in turn can be easily done in logic using the following straightforwardconsequence of the results in [4]. Lemma 12.
There exists an
FPC formula φ which given a matrix A ∈ Q I × J and a pair ofvectors b ∈ Q I , and a ∈ Q J ′ , where J ′ ⊆ J , decides if a can be extended to a solution of thelinear program Ax ≤ b . Our goal is to use Lemma 11 to show that the linear program ¯ P G can be computedwithout fixing a bijection between [ n ] and V . We define an FOC-interpretation Ψ whichtakes as input a graph G with n vertices and outputs, essentially, a relational encoding ofthe linear program ¯ P G together with the partial valuation discussed above. More precisely,Ψ outputs a matrix A ∈ Q I × J and a pair of vectors b ∈ Q I , and a ∈ Q J ′ , where J ′ ⊆ J ,such that a can be extended to a solution of Ax ≤ b if and only if G has the property ofinterest. To encode the fact that J ′ ⊆ J we introduce an extra binary relation symbol F of type ¯ J ′ × ¯ J for an injective function from the index set J ′ to the index set J . Hence, Ψoutputs a structure over the vocabulary L vec , . ∪ L vec , . ∪ L vec , . ∪{ F } modified, for simplicity,in such a way that there is a single sort symbol ¯ B for a domain of bit positions.25iven a graph G with n vertices the FOC-interpretation Ψ has access to the domain V of the graph, and the naturally ordered number domain { , . . . , n } . To represent the bitencodings of the numerical coefficients we use tuples from [ n ] k ⊆ { , . . . , n } k . Let o : [ n ] k →{ , , . . . , n k − } be the order-preserving bijection from the set [ n ] k ordered lexicographicallyto the set { , , . . . , n k − } with the natural order. For any s ∈ [ n ] k , by [ s ] we denote thenatural number o ( s ). Tuples from [ n ] k ⊆ { , . . . , n } k are also used to represent elements of Q and T . In order to do so let us fix an injective function f from Q to [ n ] k , and an injectivefunction g from T to [ n ] k . The linear program Ax ≤ b in the output of Ψ has the set ofconstraints indexed by [ n ] k × V k and the set of variables indexed by V ∪ [ n ] k × V k . Oncerestricted to the constraints indexed by f ( Q ) × V k and the variables indexed by V ∪ g ( T ) × V k it is exactly the linear program ¯ P G . All the other coefficients and constant terms in Ax ≤ b are set to 0 = ( − / z , z , z , ρ ), where z , z , z ∈ [ n ] k , and ρ is a quantifier-freeformula defining an equality type of 2 k -tuples. By T yd let us denote the set of all tuples ofthis form which satisfy one of the following conditions: • z f ( Q ) or z g ( T ), and [ z ] = 0, • z ∈ f ( Q ) and z ∈ g ( T ), and if f − ( z ) = q , g − ( z ) = t , then for every s , s ∈ [ n ] k such that the equality type of ( s , s ) is ρ , the position [ z ] in the binary encoding ofthe denominator of the coefficient of the variable indexed by ( t, s ) in the constraintindexed by ( q, s ) in ¯ P carries the 1-bit.It follows from Lemma 11 that the set T yd carries all information about the denominators ofthe coefficients of the auxiliary variables in ¯ P .Similarly, we define sets T ys , T yn , T xs , T xn , T xd , and C s , C n , C d to carry all the informationabout the signs and the bits of the numerators and the denominators of: the coefficientsof the auxiliary variables, the coefficients of the variables in { x ij } ≤ i,j ≤ n , and the constantterms, respectively.Given a graph G with the set of vertices V of size n and the set of edges E the interpre-tation Ψ does the following:1. defines the domain of ¯ I as [ n ] k × V k , the domain of ¯ J as V ∪ [ n ] k × V k , the domain of ¯ J ′ as V , and the domain of ¯ B as [ n ] k , where V is the domain of G , and [ n ] ⊆ { , . . . , n } is a subset of the number domain,2. defines the relation ≤ for the linear order on ¯ B as the lexicographic order with respectto the natural order of the number domain,3. defines the relation F of type ¯ J ′ × ¯ J as the equality relation on V ,4. defines the ternary relation P A d of type ¯ I × ¯ J × ¯ B for encoding the denominators ofthe entries of the matrix A as a union of two relations. The first is defined as a subsetof ([ n ] k × V k ) × ([ n ] k × V k ) × [ n ] k consisting of tuples ( s , v , s , v , s ) for which thereexists ( z , z , z , ρ ) in T yd such that the tuple ( v , v ) satisfies ρ , and for every i ∈ [3]it holds s i = z i . The second is defined as a subset of ([ n ] k × V k ) × V × [ n ] k consisting26f tuples ( s , v , v, w, s ) for which there exists ( z , z , ρ ) in T xd such that the tuple( v , v, w ) satisfies ρ and s = z , and s = z ,5. defines the relations P A s , P A n , P b s , P b n , P b d in a similar way as P A n ,6. defines the binary relations P a s , P a n , P a d of type ¯ J ′ × ¯ B for encoding the entries of thevector a in the following way: the entries (( v, w ) , s ) of P a s , P a n , P a d are defined to encode1 = ( − / − / v, w ) ∈ E or not.Note that by existential quantification over the sets T yd and T xd we really mean a disjunction.And by s i = z i we mean the 2-variable FO-formula of size O ( kn ) which, for every j ∈ [ k ],says that the j -th component s i,j of s i is the z i,j -th component of [ n ], using the order on thenumber domain. Observe also that Ψ, as described, is not rigorously an FOC-interpretation,but it is not difficult to see that it can be easily turned into such.The interpretation Ψ has O ( k ) variables. Its size is polynomial in n k , in k , and in thenumber of equality types of 2 k tuples, that is, polynomial in n k , k , and (2 k ) k . Since in ourcase k = O ( n ), the size of Ψ is simply n O ( k ) .Now by composing Ψ with the FPC formula φ from Lemma 12 we obtain an FPC formula ψ which given a graph G with n vertices decides if G has the property of interest. The formula ψ has l = O ( k ) variables and size n O ( k ) . We translate it into a formula θ of C l such that ψ is equivalent to θ on all structures of size at most n and θ is of size polynomial in the size of ψ , in l , and in n l (cf. Subsection 2.2). Hence, in terms of k and n , the formula θ has O ( k )variables and size n O ( k ) .We have therefore shown that a property of graphs with n vertices recognized by a graph-symmetric polytope lift of size s is defined by a C k formula, where k = O (log( s ) / (log( n ) − log log( s ))). Moreover, if s is at most weakly exponential, then for some positive real ǫ wehave k = O (log( s ) / (log( n ) − log log( s ))) = O (log( s ) / ( ǫ log( n ))) = O (log( s ) / log( n )). Hence,in this case the size of θ is n O ( k ) = s O (1) . This finishes the proof of Lemma 5 and this section. In this section we develop the main consequences of our results. We start by establishingthe main theorem of the paper, which characterizes the expressive power of symmetric linearprograms. We continue with the applications to upper and lower bounds. And end with theobservation that for random graphs over appropriate distributions symmetric LP lifts are aspowerful as general Boolean circuits. If C is a class of finite L -structures of some single-sorted vocabulary L , and n is a positiveinteger, we write C n for the set of all structures in C of cardinality n . We write s C ( n ) forthe size of a smallest L -symmetric Boolean circuit that recognizes C n , and lp C ( n ) for thesize of a smallest L -symmetric LP lift that recognizes C n . Similarly, we write w C ( n ) for the counting-width of C n , i.e., the smallest number of variables k of a C k -formula that defines C n L -structures of cardinality n , and sw C ( n ) for the counting size-width of C n , i.e., thesmallest k such that there is a C k -formula of size at most n k that defines C n on L -structuresof cardinality n . Theorem 1.
Let C be a class of finite L -structures of some vocabulary L . If lp C ( n ) is atmost weakly exponential, then1. s C ( n ) Ω(1) ≤ lp C ( n ) ≤ s C ( n ) O (1) ,2. Ω(sw C ( n )) ≤ log(lp C ( n )) / log( n ) ≤ O (sw C ( n )) .Proof. The upper bound in is a direct consequence of Lemma 1, and this holds with-out any assumption on the growth rate of lp C ( n ). The lower bound in follows fromLemma 5: Write s = lp C ( n ) and choose k = c log( s ) / (log( n ) − log log( s )) for a large c tobe specified later. Since lp C ( n ) is at most weakly exponential we have s ≤ n − ǫ for some ǫ > n . Hence k = O (log( s ) / log( n )) with the hidden constant in thebig-oh notation dependent on ǫ as in 1 /ǫ . Now, for the appropriate constant in the big-ohin k = O (log( s ) / log( n )), Lemma 5 says that there is a C k -formula that defines C and hassize polynomial in s , since again lp C ( n ) is at most weakly exponential. If the constant inthe big-oh in k = O (log( s ) / log( n )) is chosen big enough, we get that the size polynomialin s is bounded even by n k , so sw C ( n ) = O (log( s ) / log( n )) as was to be proved. In turn,these two imply the lower bound in and the upper bound in through the well-knownrelationship s C ( n ) ≤ n O (sw C ( n )) (see [33]). By Theorem 1, any class of graphs of unbounded counting width cannot be recognized bypolynomial-size symmetric LP lifts. More strongly, in combination with the strongest knownlower bounds on counting width, Theorem 1 gives weakly exponential lower bounds of thetype 2 Ω( n − ǫ ) . We show that the strongest forms of Lemmas 1 and 5 give even larger lowerbounds. Lower bounds on symmetric lifts and circuits
In the sequel, let 3-XOR refer to theconstraint satisfaction problem of deciding whether a system of 3-variable parity constraintson { , } -valued variables is satisfiable, and let 3-SAT refer to the satisfiability problem for 3-CNF formulas. In both cases, an instance is presented as a finite structure that encodes theincidence structure of the constraints: the domain is the disjoint union of the set of variablesand the set of constraints, and the relations carry one monadic relation for each type ofconstraint that indicates which constraints are of that type, and three binary relations thatindicate the three variables that participate in each constraint. Note that the instances forthese problems are not plain graphs but graphs with coloured vertices and edges. Theorem 2.
Every graph-symmetric LP lift or Boolean threshold circuit that recognizes theclass of Hamiltonian graphs with n vertices, or the class of 3-colourable graphs with n vertices,or the class of satisfiable 3-SAT instances with n variables, or the class of satisfiable 3-XOR nstances with n variables, has size Ω( n ) . Moreover, for 3-colouring, 3-SAT, and 3-XOR, thelower bound holds even on the class of instances with O ( n ) edges, O ( n ) clauses, and O ( n ) constraints, respectively. Before we enter the proof let us note that these 2 Ω( n ) lower bounds for 3-colouring, 3-XORand 3-SAT are optimal up to the multiplicative constant in the exponent. We discuss thislater in this section. Now we turn to the proof of Theorem 2. First we handle 3-colourability,then Hamiltonicity. As intermediate steps towards both we do 3-XOR and 3-SAT.By Lemma 5, for obtaining the lower bound for LP lifts it suffices to show that any C k -sentence that defines the class of n -vertex 3-colourable graphs has k = Ω( n ): indeed, when-ever s ≤ n/d , we have log( s ) / (log( n ) − log log( s )) ≤ n/ ( d log( d )) . (26)By Lemma 1, the claim then follows for Boolean threshold circuits. A result from theliterature that is quite close to the k = Ω( n ) that we need can be found in Section 4.2in [13], but the analysis in there gives k = Ω( √ n ), and not k = Ω( n ). While it shouldbe possible to modify the construction in [13] to get what we need, we refer to a morerecent construction that achieves what we want for the problems 3-XOR and 3-SAT, andthen proceed by reduction. These intermediate steps will also be useful when we discussHamiltonicity. Theorem 3 (see Theorem 3.7 and 3.8 in [5] and Lemmas 22 and 23 in [15]) . There exist c, d > such that, for every k and every sufficiently large n , every C k -sentence that separatesthe class of satisfiable 3-XOR (resp. 3-SAT) instances with n variables and cn constraintsfrom the class of unsatisfiable ones has k ≥ dn . Neither [5] nor [15] state the linear bound cn on the number of constraints, but it easilyfollow from both proofs. Concretely, it follows from Lemma 3.3 in [5], in which the bound is stated. Now we proceed by reduction in order to get the same result for 3-colouring: Lemma 13.
There exist c, d > such that, for every k and every sufficiently large n ,every C k -sentence that separates the class of 3-colourable graphs with n vertices and cn edgesfrom the class of non-3-colourable ones has k ≥ dn .Proof. In the textbook reduction from 3-SAT to the problem of deciding whether a graph is 3-colourable (see, e.g., [34]), the output graph has one gadget with two vertices for each variablein the input formula, one gadget with six vertices for each constraint of the input formula(the reduction in [34] uses only three vertices for each constraint because the reduction startsat NAE-SAT; starting at 3-SAT we need six vertices), and one special vertex. The edgesare local to each gadget, plus two edges from each variable gadget to the special vertex, anda constant number of edges for each variable occurrence in the input formula between theconstraint gadget where the variable appears, and the corresponding variable gadget. Thereare no other vertices or edges in the graph. It is clear from the construction that if the inputformula has n vertices and m constraints, then this graph has O ( n ) + O ( m ) vertices and29 ( n ) + O ( m ) + O ( m ) edges, since each of the m constraints contributes three occurrences.And it is not difficult to see that any C k formula that separates the 3-colourable graphs thatare output by the reduction from the non-3-colourable ones can be converted into a C O ( k ) formula that separate the satisfiable 3-SAT instances that are input to the reduction fromthe unsatisfiable ones. Another way to see this is by noting that the reduction is definable bya uniform quantifer-free interpretation (without the need for any ordering, or parameters, onthe input structure), from which the claim on C O ( k ) -definability follows from the closure ofthe logic under quantifier-free interpretations (see Lemma 2.1 in [5]). Now the claim followsfrom Theorem 3.For Hamiltonicity we follow the same path. A well-known result of Dahlhaus [12] gives afirst-order definable reduction from 3-SAT to Hamiltonicity that does not require any linearorder on the input. However, that reduction is quadratic and would only achieve a lowerbound of the form k = Ω( √ n ) for n -vertex graphs. We work out a linear reduction: Lemma 14.
There exists d > such that, for every k and every sufficiently large n , every C k -formula that defines the class of Hamiltonian graphs with n vertices has k ≥ dn .Proof. In this case we need a minor modification of the textbook reduction from 3-SAT to theproblem of deciding whether a graph is Hamiltonian in, e.g., [34]. As in the 3-colouring case,the output graph of the textbook reduction has one constant-size gadget for each constraint,and also one constant-size gadget for each variable. There is also one constant-size gadget foreach variable-occurrence in the input 3-CNF formula, and three additional special vertices.This defines all the vertices of the graph. See Figure 9.7 of [34]. Since each clause contributesthree variable occurrences, the total number of vertices is O ( n ) + O ( m ) + O ( m ), where n isthe number of variables, and m is the number of clauses of the input formula. However, fordefining the edges of this reduction, one needs a linear order on the variables of the inputformula for creating the path of variable gadgets in Figure 9.7 of [34]. Such a linear order isnot available in our encoding (and cannot be available as otherwise the proof of Lemma 13breaks down). Here is where we modify the construction. Instead of aligning the variablegadgets in the form of a path, we arrange them in the form of a big clique as in Figure 1.This modification does not introduce any new vertices and does not depend on any givenordering of the variables in the input formula. The analysis that proves that the reductionis correct is the same as in [34]. As in the proof of Lemma 13, it is not hard to see thatthis transformation is definable by a uniform quantifier-free interpretation, still withoutparameters or linear orders, and the result follows again from Theorem 3 and the closure ofthe logic under quantifier-free interpretations. Proof of Theorem 2.
Apply Theorem 3 and Lemmas 13 and 14 to Lemma 5, for LP lifts,and then to Lemma 1, for circuits, and in both cases use equation (26).
Lower Bound on the TSP Polytope
As stated in the introduction, Yannakakis provedthat the travelling salesman polytope does not have subexponential symmetric LP lifts. Here30igure 1: The variable gadgets arranged in the form of a big (unordered) clique. All stripedvertices are connected by a big clique. The rightmost and leftmost non-striped verticesdenote the entry and exit points, respectively, and they are also connected to all the stripedvertices. These gadgets are connected with the rest of the graph as in Figure 9.7 of [34].we show that this same lower bound follows from Theorem 2. In the next section we will seethat the same type of argument cannot work for the matching polytope.If G = ( V, E ) is a graph with V = [ n ], the incidence vector of G is x G = ( x Gij : i, j ∈ [ n ]),the vector in R [ n ] defined by x Gij = 1 if ( i, j ) is an edge of G , and x Gij = 0 if ( i, j ) is not anedge of G . Let TSP n denote the convex hull of all the vectors of the form x C , where C is aHamilton cycle of the complete graph K n on n vertices. We want to show that every graph-symmetric LP lift that has TSP n as shadow must be of exponential size as a consequence ofTheorem 2. Theorem 4 (Theorem 2 in [40]) . Every graph-symmetric LP lift that has
TSP n as shadowhas size Ω( n ) .Proof. By Theorem 2, it suffices to show that if TSP n were the shadow of a subexponential-size graph-symmetric LP lift, then there would be a subexponential-size graph-symmetric LPlift that separates the x G for which G is Hamiltonian from the x G for which G is notHamiltonian. Assume then that P were symmetric LP lift of size 2 o ( n ) whose shadow isTSP n ; let us say that its principal variables are y = ( y ij : i, j ∈ [ n ]) and that its auxiliaryvariables are z = ( z k : 1 ≤ k ≤ p ( n )). Consider the following LP on the x ij , y ij and z k variables: 0 ≤ y ij ≤ x ij for each i, j ∈ [ n ]( y , z ) ∈ P. (27)Since by assumption P is graph-symmetric, Q is also graph-symmetric. Hence, it sufficesto show that the projection of Q on the x -variables separates Hamiltonian graphs fromnon-Hamiltonian graphs.It is clear that if G is a graph that contains a Hamilton cycle C , then x G is in theprojection of Q on the x -variables: choose y = x C , and let z witness its membership in P .Conversely, assume that G is a graph and that x G is in the projection of Q on the x -variables. Then there exists y ∗ = ( y ∗ ij ) ij that is in the TSP polytope and satisfies theinequalities 0 ≤ y ∗ ij ≤ x Gij for each i, j ∈ [ n ]. This means that the support of y ∗ defines a31ubgraph of G , and at the same time that y ∗ is a convex combination of Hamilton cyclesof K n . Let y C be one of the vectors in this convex combination, where C is a Hamilton cycleof K n . The support of y C is of course included in the support of y ∗ , which means that C isalso a subgraph of G . So G contains a Hamilton cycle and is thus Hamiltonian. Upper bounds
Let us start with the simple observation that the lower bounds of type 2 Ω( n ) on 3-colouring, 3-SAT and 3-XOR are optimal: all three cases can be solved by symmetricBoolean threshold circuits of size 2 O ( n ) , and hence by symmetric LP lifts of size 2 O ( n ) byLemma 1. Here n is the number of vertices or variables, respectively. This follows fromthe fact that all three problems are definable in the existential fragment of monadic second-order logic (i.e., monadic NP), which on structures of size n , straightforwardly translate intosymmetric Boolean circuits of size 2 O ( n ) . For Hamiltonicity, the straightforward symmetricupper bound is only 2 O ( n log n ) on n -vertex graphs. It looks plausible that Bellman [9] andHeld-Karp [22] dynamic programming O ( n n ) algorithm [9, 22] could be implemented in asymmetric Boolean circuit, but we are not aware of a reference where this has been workedout. What is known is that Hamiltonicity is not definable in (full) monadic second-orderlogic (see [17]).The list of problems for which we proved a lower bound in Theorem 2 includes one thatis decidable in polynomial time, i.e., 3-XOR. This means that any polynomial-size familyof LP lifts or threshold circuits that recognizes 3-XOR must a fortiori be asymmetric. Onthe other hand, Theorem 1 says that any problem that is definable in FPC has polynomial-size symmetric LP-lifts. This includes graph planarity [20], any polynomial-time decidableproperty of graphs that exclude some minor [21], matrix singularity over rationals [10],solving systems of linear equations over rationals [25], and many others. By the resultsin [4], it also includes the problem of deciding whether a (general, not necessarily bipartite)graph contains a perfect matching. The family can even be taken to be polynomial-timeuniform by applying the construction of Lemma 1 to the “easy half” of the equivalencebetween FPC and polynomial-size symmetric threshold circuits in [3]. Corollary 2.
There is a (polynomial-time uniform) family of graph-symmetric LP lifts ofpolynomial size that recognizes the class of graphs that have a perfect matching.
This should be contrasted with the fact, proved by Yannakakis, that any symmetric LPlift of the perfect matching polytope PM n has size 2 Ω( n ) . Here, PM n is defined as the convexhull of all vectors of the form x M , where M is the edge set of a perfect matching of thecomplete graph K n on 2 n vertices. Capturing PM n by an LP lift or recognizing the classof graphs that have a perfect matching by an LP lift are different tasks. Both objects couldbe used for deciding whether a given graph has a perfect matching, but capturing PM n hasa demanding structural requirement that has no analogue in the other task. The upperbound of Corollary 2 also means that the argument that was used for deriving lower boundsfor TSP n in the proof of Theorem 4 cannot be adapted to PM n . Indeed, we do not knowwhether there is any route at all for deriving lower bounds for PM n via our results.In view of Corollary 2 one may wonder whether the convex hull of the incidence vectors ofgraphs that have a perfect matching has a small symmetric LP lift. This, however, is easily32een to not be the case: if it had, then its intersection with the halfplane P i,j ∈ [2 n ]: i = j x ij = 2 n would be a small symmetric LP lift for PM n . Let G ( n, p ) denote the Erd˝os-R´enyi distribution on n -vertex labelled graphs with edge prob-ability p . We write G ∼ G ( n, p ) to mean that G is a random graph distributed as in G ( n, p ).In this section we argue that, for average-case problems with respect to the uniform distri-bution G ( n, / G ( n, /
2) fromsome other distribution, polynomial-size symmetric LPs are as powerful as arbitrary notnecessarily symmetric Boolean circuits. For average-case problems, this is indeed a directconsequence of our main result and the following well-known fact in descriptive complexitytheory:
Theorem 5 (Corollary 4.8 in [23]) . For every polynomial-time decidable class of graphs C there is an FPC -definable class of graphs C ′ for which the probability that a random graph G ∼ G ( n, / falls in the symmetric difference C ∆ C ′ is o (1) . The point of Theorem 5 is that the FPC formula that defines C ′ does not require anyorder on the input graph, hence our Theorem 1 applies. Theorem 5 is indeed a consequenceof the Immerman-Vardi Theorem [27, 38] and the fact that a linear order is, asymptoticallyalmost surely on G ( n, / G ( n, /
2) from someother distribution of random graphs, to which a direct application of Theorem 5 does notlook possible.We focus the discussion on the planted clique problem since it is one of the best studiedsuch problems, although it will be clear from the discussion that the phenomenon is moregeneral. Let G ( n, p, k ) denote the distribution that results from drawing a random graphfrom G ( n, p ) and then planting a random k -clique in it, i.e., adding the edges of a k -cliqueon a uniformly chosen subset of k vertices. Following [8], the planted clique problem, alsoknown as the hidden clique problem, comes in three flavours. Informally stated, these are: • search: given G ∼ G ( n, p, k ), find the planted clique, • refutation: given G ∼ G ( n, p ), certify that the clique number is less than k , • decision: given G ∼ G ( n, p ) or G ∼ G ( n, p, k ), determine which is the case.Formally, the decision version can be stated as follows. We say that a class of graphs C solvesthe decision version of the planted clique problem with parameters p = p ( n ) and k = k ( n )and advantage ǫ = ǫ ( n ) > n the following hold:1. if G ∼ G ( n, p ), then G is in C with probability at least 1 / ǫ ,2. if G ∼ G ( n, p, k ), then G is in C with probability at most 1 / − ǫ .33olvable in polynomial time (or in FPC, or by a family of LP-lifts, etc.) means that it issolvable by a class of graphs C that can be recognized in polynomial time (or in FPC, orby a family of LP-lifts, etc.). It should be clear that if the decision version is hard, then theother two versions of the problem can only be harder.The planted clique problem is an easy-to-state average case problem that has a longhistory. Its formulation is attributed to Jerrum [29] and Kuˇcera [32], independently. In therange k ( n ) = ω ( √ n log n ), a simple algorithm based on degree sequences solves the searchproblem in polynomial time [32]. In the range k ( n ) = Ω( √ n ), a spectral based algorithmis known to solve the problem in polynomial time [2], but spectral-free algorithms are alsoknown that run even in linear time [18]. For k ( n ) = o ( √ n ) the status of the problem isfamously open, even for its decision variant. Some lower bounds are known in restrictedmodels, such as the original lower bound for Markov chain methods in [29]. Lower boundsare also known for (symmetric) linear and semidefinite program formulations of the problem.We discuss these next.We formulate the clique problem on a graph G = ( V, E ) as a non-linear polynomialoptimization problem: For each vertex v ∈ V introduce one variable y v that stands for theindicator whether v belongs to the clique. The clique number is the maximum of P v ∈ V y v subject to the constraints that y u y v = 0 for each non-edge ( u, v ) E , and y v − y v = 0for each u ∈ V . For the refutation and the decision versions of the problem, it is morenatural to turn the objective function into a constraint P v ∈ V y v ≥ k . This gives a differentquadratic program feasibility problem for each graph G . Thinking of G as given by the { , } -vector ( x u,v ) u,v ∈ [ n ] in the usual way, the programs can be made uniform, i.e., a singlequadratic program feasibility problem serves all graphs with V = [ n ]: P v ∈ [ n ] y v ≥ ky u y v ≤ x u,v for u, v ∈ [ n ] y v − y v = 0 for v ∈ [ n ]While this a hard-to-solve quadratic program, there are several tractable relaxations thatone can study. Two systematic methods for generating such relaxations were introduced byLov´asz and Schrijver in [31], and Sherali and Adams in [37]. Both cases start by (naively) lin-earizing the quadratic program, i.e., replacing each quadratic term y u y v in the constraintsby a new variable y { u,v } . The result is a very weak symmetric LP relaxations with auxiliaryvariables, but this is only the first level of the LS and SA hierarchies. The successive levelsof the hierarchies are obtained by adding linear constraints that can be proved valid for theconvex hull of solutions of the quadratic program by iterated applications of a simple rule ofinference.It can be seen that the levels of the hierarchies are always graph-symmetric LPs, withthe d -th level having n O ( d ) auxiliary variables and constraints, where n = | V | . The strengthsof the relaxations converge to the quadratic program in the sense that the polytopes theyproject to are tighter and tighter approximations of the convex hull of solutions of thequadratic program. Moreover, the exact convex hull is eventually reached no later thanthe n -th level. 34n order to appreciate the symmetry of the LPs that define the sucessive levels of thehierarchy, it useful to enter the details for the definition of the SA hierarchy. The firststep in producing the d -th level of the SA hierarchy is to formally multiply each constraintof the quadratic program by an extended monomial of the form Q w ∈ I y w Q w ∈ J (1 − y w )for I, J ⊆ V with I ∩ J = ∅ and | I ∪ J | ≤ d −
1. The second step is to expand out the formalexpressions into sums of degree d + 1 monomials, and multilinearize every monomial on the y -variables; note that this is a valid step only under the constraint y v − y v = 0, hence for { , } -assignments. In the third step we introduce a new variable y I for each I ⊆ V with | I | ≤ d + 1, and a new variable y u,v,I for each u, v ∈ V and each I ⊆ V with | I | ≤ d − Q w ∈ I y w by the new variable y I , and each monomial x u,v Q w ∈ I y w by the new variable y u,v,I . The link between the twotypes of variables is established by setting x u,v = y u,v, ∅ and y ∅ = 1. The symmetry of theresulting LP is obvious since the image I ( π ) of a set I ⊆ V by a permutation π ∈ Sym V has the same cardinality as I . The result is a graph-symmetric LP with n O ( d ) auxiliaryvariables and constraints, whose shadow eventually captures the convex hull of solutions ofthe quadratic program. Any level that approximates this hull well enough, even on averageon G ( n, / d , the d -th level of the LS hier-archy cannot distinguish between G ( n, /
2) and G ( n, / , k ( n )) when k ( n ) = o ( √ n ) withany significant advantage. Their lower bound is much stronger than we stated it since itapplies even to LS + , the semi-definite programming variant of the LS-hierarchy. For the SAhierarchy, the same result is attributed to the folklore, and more recent works [7, 26] have ob-tained analogous results for the even stronger Lasserre/Sums-of-Squares (SOS) hierarchies,also proving that the problem stays hard for their constant d level when k ( n ) = o ( √ n ). Fur-ther, in certain contexts, it is possible to prove that the Sherali-Adams hierarchy is optimal among symmetric LP lifts of comparable size. This includes a model of LP lifts for solvingBoolean constraint satisfaction problems (see Theorem 4.1 in [11]). This means that, insuch restricted contexts, proving lower bounds on the levels of Sherali-Adams is enough forgetting size lower bounds for any symmetric LP lift.In view of such success in proving lower bounds on the size of symmetric LP lifts, startingwith Yannakakis, and including the discussion above on hierarchies for the planted cliqueproblem, and also given our own lower bounds from Section 5.2, the following consequenceof Theorem 1 may come as a surprise: Corollary 3.
If the planted clique problem with parameters p = 1 / and k = k ( n ) is solvablein polynomial time with advantage ǫ > , then it is also solvable by a (polynomial-timeuniform) family of polynomial-size graph-symmetric LP lifts with advantage ǫ − o (1) . In the rest of this section we show how to derive the descriptive complexity versionof Corollary 3, from which Corollary 3 follows at once from Theorem 1. The descriptivecomplexity variant relies on the following well-known fact, which builds on the almost suregraph canonization methods of [6]: 35 heorem 6 (Theorem 4.6 in [23]) . There is an
FOC -formula φ ( x, y ) such that, for all ǫ > and all sufficiently large n , if G ∼ G ( n, / , then φ ( x, y ) defines a strict linear order on thevertices of G with probability at least − ǫ . We note that Theorem 6 is one of the two ingredients in the proof of Theorem 5; thesecond one is the Immerman-Vardi Theorem. While our proof uses only these two ingredientstoo, we do not see a direct way of getting it from Theorem 5.
Theorem 7.
If the planted clique problem with parameters p = 1 / and k = k ( n ) is solvablein polynomial time with advantage ǫ > , then it is solvable in FPC with advantage ǫ − o (1) .Proof. Suppose that C is a polynomial time decidable class of graphs that solves the problemwith advantage ǫ >
0. Let C ′ denote the class of all ordered expansions of graphs in C , i.e.,the structures in C ′ are finite structures over the vocabulary L = { E, R } , where E and R arebinary relations symbols, and E is interpreted as the edge relation of some graph in C , and R is interpreted as a strict linear order on its set of vertices. By the Immerman-Vardi Theorem,there is an FP formula ψ that defines C ′ on the class of all ordered graphs. Let φ ( x, y ) bethe FOC formula from Theorem 6, and let θ be the conjunction of the following sentences: ∀ x ( ¬ φ ( x, x )) ∀ x ∀ y ∀ z ( ¬ φ ( x, y ) ∨ ¬ φ ( y, z ) ∨ φ ( x, z )) ∀ x ∀ y ( x = y ∨ φ ( x, y ) ∨ φ ( y, x )) ψ [ R/φ ] . This sentence says that φ defines a strict linear order and ψ holds when each occurrenceof R is replaced by φ . This is an FPC formula over the vocabulary of unordered graphs thatdefines a class D of graphs. We claim that D solves the problem with advantage ǫ − o (1).By assumption and the fact that C ′ contains all ordered expansions of graphs in C we have that the probability that some and hence every ordered expansion of G satisfies ψ is at least 1 / ǫ when G ∼ G ( n, / / − ǫ when G ∼ G ( n, / , k ).Now, if G ∼ G ( n, / φ does not define a linear order is o (1),and the probability that some and hence every ordered expansion of G satisfies ψ is atleast 1 / ǫ , so the probability that G satisfies θ is at least 1 / ǫ − o (1). On the otherhand, if G ∼ G ( n, / , k ), then the probability that some and hence every ordered expansionof G satisfies ψ is at most 1 / − ǫ , so the probability that G satisfies θ is even smaller, and1 / − ǫ ≤ / − ǫ + o (1). It follows that D solves the problem with advantange ǫ − o (1). Our main result Theorem 1 establishes a tight three-way correspondence between symmetricBoolean threshold circuits, symmetric LP lifts, and bounded-variable formulas of countinglogic. We used this to derive upper and lower bounds on the size of symmetric LPs and36ymmetric Boolean threshold circuits. We also used it to bound the asymmetric circuit com-plexity of the planted-clique problem by its symmetric LP lift complexity, up to a polynomialfactor. There are several directions for further investigation that are suggested by this work.The first one concerns the problem of circuits vs formulas . Composing the first inequalityin the first item of Theorem 1 with the second inequality in the second item of Theorem 1,we get the result that, for every constant k , every symmetric Boolean threshold circuit of sizeat most n k translates into an equivalent C O ( k ) -formula of size n O ( k ) . This is a size-efficienttranslation from circuits into formulas, albeit of different types: the source is a Booleanthreshold circuit and the target is a counting logic formula. An explicit and uniform suchtranslation is given in [3], and similarly, AND-OR-NOT circuits can translate into familiesof L k -formulas, where L k stands for the k -variable fragment of first-order logic, withoutcounting.At first sight, the translation from circuits to formulas is unexpected as the circuit valueproblem is P-complete, while the formula value problem is in NC . However, as we noted,these are not Boolean formulas and the natural translation of formulas of C k or L k intocircuits necessarily yields circuits with high fan-out. This also accords with the well-knownfact that the combined complexity of the logic L k is P-complete for each k ≥ k (or C k )? Could such a translation shed lighton the descriptive complexity of NC ? On the NC vs P question? Similar questions canbe asked for symmetric bounded-depth threshold circuits, and other classes of circuits. Itis worth noting that both Theorem 5 and Theorem 7 on the planted-clique problem scaleall the way down to uniform TC and FOC. This is so because Theorem 6 gives a FOC-formula and, in the presence of a linear order on the input, FOC captures uniform TC (seeProposition 12.6 in [28]). We view all this as motivation for finding a symmetric LP modelof symmetric bounded-depth threshold circuits.A different line of research that is suggested by our work relates to the computationalcomplexity of the linear programming feasibility problem. It is well-known that this problemis P-complete under logspace reductions and a question posed in [4], where it was shown thatthis problem is in FPC, is whether it is complete for FPC under (say) FOC reductions. Ourproof of Lemma 1 gives one route towards answering this question. The construction inthe proof of the lemma should yield a first-order interpretation from the threshold circuitvalue problem to the LP feasibility problem. Since the known translations from FPC intosymmetric threshold circuits are first-order interpretations themselves, the result shouldfollow by composition. The FPC-completeness of the LP feasibility problem was also provedby Pakusa (unpublished manuscript), independently of our work. Pakusa’s constructionis very different from ours and raises the question whether the construction in his proofcould yield a different proof of Lemma 1. His construction is inspired by the Sherali-Adamshierarchy of LP relaxations [37] (discussed in Section 5) and could well provide a moreprincipled method than ours for constructing LP lifts with useful properties.37 cknowledgments First author partially funded by European Research Council (ERC) un-der the European Union’s Horizon 2020 research and innovation programme, grant agreementERC-2014-CoG 648276 (AUTAR) and MICCIN grant TIN2016-76573-C2-1P (TASSAT3).The second author was partially supported by a Fellowship of the Alan Turing Instituteunder the EPSRC grant EP/N510129/1. The third author funded by the European Union’sHorizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grantagreement No 795936. Some of the work reported here was initiated at the Simons Institutefor the Theory of Computing during the programme on Logical Structures in Computationin autumn 2016. We are grateful to Matthew Anderson for some very stimulating discussionson this topic. In particular, he suggested using Lemma 9 to prove the existence of supports.We also thank Wied Pakusa for sharing his manuscript on the FPC-completeness of linearprogramming with us.
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