A note on VNP-completeness and border complexity
AA note on VNP-completeness and bordercomplexity
Christian Ikenmeyer
University of [email protected]
Abhiroop Sanyal
Chennai Mathematical [email protected]
Abstract
In 1979 Valiant introduced the complexity class VNP of p-definable families of polynomials, hedefined the reduction notion known as p-projection and he proved that the permanent polynomialand the Hamiltonian cycle polynomial are VNP-complete under p-projections.In 2001 Mulmuley and Sohoni (and independently Bürgisser) introduced the notion of bordercomplexity to the study of the algebraic complexity of polynomials. In this algebraic machine model,instead of insisting on exact computation, approximations are allowed. In this short note we studythe set VNPC of VNP-complete polynomials. We show that the complement VNP \ VNPC liesdense in VNP. Quite surprisingly, we also prove that VNPC lies dense in VNP. We prove analogousstatements for the complexity classes VF, VBP, and VP.The density of VNP \ VNPC holds for several different reduction notions: p-projections, borderp-projections, c-reductions, and border c-reductions. We compare the relationship of the VNP-completeness notion under these reductions and separate most of the corresponding sets. Borderreduction notions were introduced by Bringmann, Ikenmeyer, and Zuiddam (JACM 2018). Ourpaper is the first structured study of border reduction notions.
Theory of computation → Algebraic complexity theory
Keywords and phrases algebraic complexity theory, VNP, border complexity, reductions, complete-ness, topology
Funding
Christian Ikenmeyer : CI supported by DFG grant IK 116/2-1
Acknowledgements
We thank Michael Forbes for helpful insights about [13]. Moreover, Theorem 1was initially only stated for VNP and we thank an anonymous reviewer for noting that it works inhigher generality. We thank Josh Grochow for discussions about border complexity and topologythat led to significant adjustments in Section 2.2 and the simplification of some proofs.
Valiant’s famous determinant versus permanent conjecture [23] states that the algebraiccomplexity class VBP (polynomials that can be written as determinants of polynomiallylarge matrices of linear polynomials) is strictly contained in the class VNP (polynomialsthat can be written as Hamilton cycle polynomials of polynomially large matrices of linearpolynomials, see Section 2.1). In 2001 Mulmuley and Sohoni in their Geometric ComplexityTheory approach towards resolving Valiant’s conjecture [20] stated a strengthening of theconjecture (VNP VBP) that is based on border complexity , which was stated independentlyfor circuits by Bürgisser [8, hypothesis (12)] (VNP VP). The advantage of working with theclosures of complexity classes is that this makes a large set of tools from algebraic geometry a r X i v : . [ c s . CC ] F e b A note on VNP-completeness and border complexity
VFVBPVPVNPCVNP
Figure 1
The known inclusions of the classical algebraic complexity classes. VF is the class offamilies of polynomials with polynomially sized formulas, VBP is the class of families of polynomialsthat can be written as polynomially large determinants of matrices of linear polynomials, VP is theclass of families of polynomials with polynomially sized circuits. VNPC is the set of VNP-completefamilies. From a topological perspective such a depiction can be misleading, because VNPC liesdense in VNP and also VNP \ VNPC lies dense in VNP (under p -projections), see Theorem 1. and representation theory available, see e.g. [4]. The hope is that VBP and VNP can still beseparated in this coarser setting. Indeed, it is a major open question in geometric complexitytheory whether or not VBP = VBP, see [10]. If VBP = VBP, then Valiant’s conjecture mustin principle be provable by algebraic geometry, provided it is true. If VNP ⊆ VBP, then theGeometric Complexity Theory approach fails unsalvageably, while Valiant’s conjecture couldstill be true.In Boolean complexity theory the relationship between complexity classes is often depictedin diagrams. An analogue for the classical algebraic complexity classes is given in Figure 1.In this paper we see that such a depiction presents misleading topological information: Westudy the set of VNP-complete polynomials and its complement and see that surprisinglyboth lie dense in VNP, see Theorem 1. We prove an analogous result for VF, VBP, and VP.This highlights that this topology is very coarse.We take the methods for proving Theorem 1 as a basis for studying VNP-completenessunder different reduction notions, in particular we study border- p -projections, which wererecently introduced in [6] with a focus on the border- p -projections of the iterated 2 × Fix a field F . An algebraic circuit is defined as a rooted directed acyclic graph which hasits leaf vertices labelled with variables { x , x , . . . , x n } and field constants and the internal hristian Ikenmeyer and Abhiroop Sanyal 3 nodes labelled with × (“multiplication gates”) and + (“addition gates”). By induction overthe circuit structure, each internal node computes a polynomial f ∈ F [ x , x , . . . , x n ]. Theoutput of a circuit is defined as the polynomial computed at its root. The size of an algebraiccircuit is defined as the number of nodes in the circuit.A sequence of natural numbers ( t n ) n ∈ N is called polynomially bounded if there exists apolynomial function p such that for all n ∈ N we have t n ≤ p ( n ). A sequence ( f ) = ( f n ) n ∈ N of multivariate polynomials is defined to be a p -family if the number of variables in f n andthe degree of f n are both polynomially bounded. The complexity class VP is defined as theset of all p -families that have algebraic circuits whose size is polynomially bounded. If weonly allow skew circuits, i.e., circuits for which each multiplication gate is adjacent to a leafnode, then we get the complexity class VBP. If instead we insist on the circuits to be rooted trees , then we get the complexity class VF. We have VF ⊆ VBP ⊆ VP, see e.g. [22].For fixed natural numbers N and M , a polynomial f ∈ F [ x , x , . . . , x N ] is said to bea projection of another polynomial g ∈ F [ y , y , . . . , y M ] if f = g ( α , α , . . . , α M ), where α i ∈ { x , x , . . . , x N } ∪ F . This is denoted by f ≤ g . A p -family ( f ) is said to be the p -projection of another p -family ( g ), denoted by ( f ) ≤ p ( g ), if there is a polynomially boundedfunction t : N → N such that f n ≤ g t ( n ) for all n . p -projections are the first type of reductions introduced by Valiant. Another naturalexample of reductions are c -reductions, an algebraic analogue of oracle complexity. Theoracle complexity L g ( f ) of a polynomial f with oracle g is defined as the minimum size of acircuit with + , × , and g -oracle gates (the output at these gates is the computation of thepolynomial g on the input values. The arity of a g -oracle gate equals the number of variablesin g ) that can compute the polynomial f . Consider two p -families ( f ) and ( g ). The p -family( f ) is said to be a c -reduction of ( g ), denoted by ( f ) ≤ c ( g ), if there exists a polynomiallybounded function t : N → N such that L g t ( n ) ( f n ) is polynomially bounded.Let S n denote the symmetric group on n symbols. Let C n ⊆ S n be the subset of length n cycles. The Hamiltonian cycle family (HC) is the sequence of homogeneous degree n polynomials HC n on n many variables defined via HC n := P π ∈ C n Q ni =1 x i,π ( i ) . We define the class VNP as the set of all p -families ( f ) that satisfy ( f ) ≤ p (HC). This isknown to be equivalent to ( f ) ≤ c (HC). Often a different definition is given that is closerin spirit to the counting complexity class ⊆ VNP. Valiant’s famous conjectures areVF = VNP, VBP = VNP, and VP = VNP. To find candidates outside of VF, VBP or VP,the following notion of VNP-completeness is useful.A p -family ( f ) is defined to be VNP- p -complete if ( f ) ∈ VNP and (HC) ≤ p ( f ). The setof all VNP- p -complete p -families is denoted by VNPC( ≤ p ). Analogously, a p -family ( f ) isdefined to be VNP- c -complete if ( f ) ∈ VNP and (HC) ≤ c ( f ). The set of all VNP- c -complete p -families is denoted by VNPC( ≤ c ).The main motivation behind VNP-completeness comes from the following simple obser-vation: If we would find ( g ) such that both ( g ) ∈ VNPC( ≤ p ) and ( g ) ∈ VP, then for all( f ) ∈ VNP we would have ( f ) ≤ p (HC) ≤ p ( g ), and by transitivity ( f ) ≤ p ( g ), which implies( f ) ∈ VP, thus VP = VNP. Analgously for VF and VBP instead of VP.A p -family ( f ) is defined to be VF- p -complete if ( f ) ∈ VF and ( g ) ≤ p ( f ) for every( g ) ∈ VF. The set of all VF- p -complete p -families is denoted by VFC( ≤ p ). The iterated3 × ≤ p ),see [2]. A p -family ( f ) is defined to be VBP- p -complete if ( f ) ∈ VBP and ( g ) ≤ p ( f ) forevery ( g ) ∈ VBP. The set of all VBP- p -complete p -families is denoted by VBPC( ≤ p ). The A note on VNP-completeness and border complexity determinant polynomial family is an example of an element in VBPC( ≤ p ), see [22]. A p -family ( f ) is defined to be VP- p -complete if ( f ) ∈ VP and ( g ) ≤ p ( f ) for every ( g ) ∈ VP.The set of all VP- p -complete p -families is denoted by VPC( ≤ p ). There exist specific graphhomomorphism polynomial families that are in VPC( ≤ p ), see the recent [18]. The search fornatural problems in VPC( ≤ p ) was in fact a long-standing open problem. In this section we define so-called border complexity analogues to the reduction notions inSection 2.1. This was first done explicitly in [6]. In our definitions we will use the polynomialring F [ ε ] and the field extension F ( ε ). The reduction notions from Section 2.1 are interpretedover this field extension. Therefore, even if g is a polynomial over F , we can have f ≤ g witha polynomial f over F ( ε ).A polynomial f over F is called a border projection of a polynomial g over F (denoted by f (cid:69) g ) if f + ε · h ≤ g for some polynomial h over F [ ε ]. Clearly f ≤ g implies f (cid:69) g . Weextend this definition to sequences of polynomials as follows. A sequence of polynomials ( f )is defined to be a border p -projection of another sequence of polynomials ( g ), denoted by( f ) (cid:69) p ( g ), if there is a polynomially bounded function t : N → N such that f n (cid:69) g t ( n ) forall n .The border oracle complexity L g ( f ) of a polynomial f over F with oracle access to apolynomial g over F is defined as the smallest c such that there exists a polynomial h over F [ ε ] such that L g F ( ε ) ( f + ε · h ) ≤ c , where L g F ( ε ) denotes the oracle complexity over the field F ( ε ). Clearly L g ( f ) ≤ L g ( f ) for all f and g . Consider two p -families ( f ) and ( g ). The p -family ( f ) is said to be a border c -reduction of ( g ), denoted by ( f ) (cid:69) c ( g ), if there exists apolynomially bounded function t : N → N such that L g t ( n ) ( f n ) is polynomially bounded.A p -family ( f ) is defined to be VNP-border- p -complete if ( f ) ∈ VNP and (HC) (cid:69) p ( f ).The set of all VNP-border- p -complete p -families is denoted by VNPC( (cid:69) p ). Analogously, a p -family ( f ) is defined to be VNP-border- c -complete if ( f ) ∈ VNP and (HC) (cid:69) c ( f ). Theset of all VNP-border- c -complete p -families is denoted by VNPC( (cid:69) c ).In geometric complexity theory the common type of reduction is similar to p -projectionsand border- p -projections, but instead of replacing variables with constants and variables,variables are replaced by affine linear polynomials. All proofs in this paper also work withthis version of p -projections and border- p -projections.Let C be a complexity class definable by circuits over F and F ( ε ) such as VF, VBP, VP,or VNP. A p -family ( g ) over F is defined to lie in the closure C of such a class C over F ifthere exists a function e : N → N and a p -family ( h ) over F [ ε ] such that ( h ) ∈ C ( F ( ε )) andfor every n ∈ N and i ≤ e ( n ) there exist polynomials f n,i over F with h n := g n + εf n, + · · · + ε e f n,e . The class VNPC( ≤ p ) is not defined by circuits. A p -family ( f ) over F is defined to be inVNPC( ≤ p ) if (HC) ≤ p ( f ) + ε · ( g ) for some p -family ( g ) over F [ ε ]. Analogously for ≤ c , (cid:69) p , and (cid:69) c . The class VNP \ VNPC( ≤ p ) is also not defined by circuits. A p -family ( f ) isdefined to be in VNP \ VNPC( ≤ p ) if ( f ) + ε · ( g ) ∈ VNP( F ( ε )) for some p -family ( g ) over F [ ε ] and (HC) p ( f ) + ε · ( g ). Analogously for ≤ c , (cid:69) p , and (cid:69) c , but we will not study thesethree notions in this context. The closures of VFC, VBPC, VPC, VF \ VFC, VBP \ VBPC,VP \ VPC are defined analogously by replacing VNP and using the respective completepolynomials.As noted in [6], taking the closure is not a closure operator in the usual sense of the hristian Ikenmeyer and Abhiroop Sanyal 5 definition. Still, we define a sub C ⊆ C to be dense in C if C = C . For all the sets VF,VBP, VP, VNP, it is a major open question if they are equal to their closure, see [10]. Using a fairly elementary proof we obtain the following surprising density results. (cid:73)
Theorem 1.
Let ≤ be one of ≤ p , ≤ c , (cid:69) p , or (cid:69) c . Then the set VNPC( ≤ ) is a dense subsetof VNP . Moreover, the complement
VNP \ VNPC( ≤ p ) is a dense subset of VNP . Additionally,
VFC( ≤ ) and VF \ VFC( ≤ p ) are both dense in VF ; VBPC( ≤ ) and VBP \ VBPC( ≤ p ) areboth dense in VBP ; and
VPC( ≤ ) and VP \ VPC( ≤ p ) are both dense in VP . We leave it as an open problem if VNP \ VNPC( ≤ ) is dense in VNP for the other threereduction notions. Oracle complexity is obviously too coarse to study VF-completeness,VBP-completeness, or VP-completeness.As a second result we initiate the comparison of classical and border reduction notions.We give an almost complete separation of the sets of VNP-complete polynomials under thedifferent reduction notions as follows. (cid:73) Theorem 2.
Over F ∈ { Q , R , C } we have the following diagram of inclusions (solid arrows)and non-inclusions (dashed arrows). VNPC( ≤ p ) VNPC( ≤ c )VNPC( (cid:69) p ) VNPC( (cid:69) c ) [13] Cor. 5 and Lem. 6Lem. 7 and Lem. 8 Cor. 5 and Lem. 6 The inclusions (solid arrows) are obvious. The non-inclusions are proved in the respectivelemmas as annotated in the figure.
The relative power of algebraic reduction notions has been studied before: [13] show with ashort argument thatVNPC( ≤ p ) (cid:36) VNPC( ≤ c ) . They do not study border complexity though.Border complexity has already been an object of study in algebraic complexity theory forbilinear maps since 1980 (see [3]) and is still a very active area of research today [17, 16].The study of border complexity for polynomials has recently gained significant momentum,see for example [10, 15, 6, 5]. In fact, [6] prove that VFC( ≤ p ) (cid:36) VFC( (cid:69) p ). Their proof isbased on a fairly involved analysis in [1].The Boolean world knows many types of reductions. Their relative power has beenanalyzed for example in [11, 12]. The notion of c -reductions in algebraic complexity theory isrelatively new [7]. The difference between p -projections and c -reductions plays a prominentrole in [19]. A note on VNP-completeness and border complexity
We start with an observation of [13] that we state simultaneously for VNP, VF, VBP, and VP.It is clear that the result holds in much higher generality. (cid:73)
Proposition 3 ([13]) . Fix any field ˜ F . Given a p -family ( f ) over ˜ F such that each f n hasthe following form: f n = q ( rg n + cg n ) for some p -family ( g ) over ˜ F , some fixed polynomial r over ˜ F of any degree d , some fixed polynomial q over ˜ F of even degree d that is also a perfectsquare, and some constant c ∈ ˜ F . Let s be a univariate polynomial of odd degree > d + 2 d .Then for all n : s f n . In particular for the constant family ( s ) we have ( s ) p ( f ) . Proof.
The proof is a very minor generalization of [13, Lemma 3.2]. Consider a univariatepolynomial s ( y ) of odd degree M > d + 2 d . Let deg y ( h ) denote the degree of a polynomial h in the variable y , when considered as a polynomial over the polynomial ring in all its otherconstituent variables. We claim that s cannot be written as a projection of f n , for any n . Let γ be any linear projection map. Then, deg y ( γ ( f n )) ≤ max [ deg y ( γ ( q · r · g n )) , deg y ( γ ( q · g n ))].Also, note that deg y ( γ ( q )) ≤ d and deg y ( γ ( r )) ≤ d .If deg y ( γ ( g n )) ≤ d , then deg y ( γ ( f n )) ≤ d + 2 d . If γ ( f n ) = s ( y ), this contradicts thefact that s ( y ) has degree M .Otherwise, deg y ( γ ( f n )) = deg y ( γ ( q · g n )). But q · g n is a perfect square polynomial, hencedeg y ( γ ( f n )) is even, but s ( y ) has odd degree. Hence, γ ( f n ) = s ( y ). (cid:74) Proof of Theorem 1.
First we prove that VNPC( ≤ p ) is dense in VNP. Let ( f ) ∈ VNP bearbitrary. Define h n := f n + εy (HC n − f n ) , a polynomial over F ( ε ). Let γ denote the projection map γ : y ε . We observe γ ( h n ) = HC n and hence (HC) ≤ p ( f ) + ε · ( g ) with g n = y (HC n − f n ). Therefore ( f ) ∈ VNPC( ≤ p ).The result for the other reduction types is immediate, because p -projections are the weakestnotion of reduction we consider, in particular VNPC( ≤ p ) ⊆ VNPC( ≤ c ), VNPC( ≤ p ) ⊆ VNPC( (cid:69) p ), and VNPC( ≤ p ) ⊆ VNPC( (cid:69) c ).For the other part, let ( f ) ∈ VNP be arbitrary. Define h n := f n + εf n a polynomial over F ( ε ). Obviously, ( h ) ∈ VNP( F ( ε )), but according to Proposition 3 (taking˜ F = F ( ε ), r = q = 1 and c = ε ), we have ( s ) p ( h ) for the constant family ( s ) and hence(HC) p ( h ).The analogous statements about VF, VBP, and VP that are claimed in the theorem areproved in exactly the same way by replacing (HC) by a complete polynomial family for therespective class. (cid:74) We start with a classical lemma about taking roots. (cid:73)
Lemma 4.
Suppose g = f r for some f ∈ F [ x ] of degree d and constant r , where x denotesa set of variables. Then f can be computed by a g -oracle circuit of size poly ( d ) . In particular,if g n = f rn for all n , then ( f ) ≤ c ( g ) . hristian Ikenmeyer and Abhiroop Sanyal 7 Proof. (The proof follows that of a special case of [14], the proof technique borrows from[24] and [9]). Consider a polynomial g = f r , where f has degree d . Notice that for everyinfinite field F and every nonzero polynomial f over F [ x ], there exists α ∈ F | x | , such that f ( α ) = 0. Also, shifting the variables f ( x ) f ( x + α ) is an invertible operation since youmay re-shift at the input nodes of the circuit. Thus, by appropriately shifting we may assumew.l.o.g. that g ( ) = 0. Rescaling at the output node is also an invertible operation, so wemay assume w.l.o.g. that g ( ) = 1. Then, we can write: f = (1 + ( g − /r Using the binomial theorem for rational coefficients, this gives us:(1 + ( g − /r = 1 + 1 r ( g −
1) + (cid:18) /r (cid:19) ( g − + · · · + (cid:18) /rd (cid:19) ( g − d + · · · Since g ( ) = 1, then g − x ). Thus, ( g − i has trailing monomial degree largerthan d for i ≥ d + 1. So, f = 1 + 1 r ( g −
1) + (cid:18) /r (cid:19) ( g − + · · · + (cid:18) /rd (cid:19) ( g − d mod (cid:0) x d +1 (cid:1) where x d +1 denotes the set of all monomials of degree d + 1. We have the oracle circuit for g .The modular operation can be done via Strassen’s homogenization trick [21]. Specifically,each homogeneous part can be written as a linear combination of ( d + 1) p -projections of g .Thus, computing roots using oracle gates is possible with circuits of size poly( d ). Thisproves the first part. The second part follows from the fact that p -families have polynomiallybounded degrees. (cid:74) As an immediate corollary we obtain: (cid:73)
Corollary 5. (HC ) ∈ VNPC( ≤ c ) . Proof.
By Lemma 4 we have (HC) ≤ c (HC ). Since (HC) ∈ VNPC( ≤ c ), for every ( f ) ∈ VNP we have ( f ) ≤ c (HC). By transitivity we have ( f ) ≤ c (HC ). Therefore (HC ) ∈ VNPC( ≤ c ). (cid:74)(cid:73) Lemma 6. (HC ) VNPC( (cid:69) p ) . Proof. (HC ) ∈ VNPC( (cid:69) p ) is equivalent to (HC) (cid:69) p (HC ). Since the constant p -family ( x )satisfies ( x ) (cid:69) p (HC), if we prove that ( x ) (cid:69) p (HC ), then (HC) (cid:69) p (HC ) by transitivity.Indeed, ( x ) (cid:69) p (HC ) is equivalent to the existence of a polynomially bounded function t : N → N and a p -family ( g ) over F [ ε ] such that ∀ n : x + εg n ≤ (HC t ( n ) ) . All projectionsof (HC t ( n ) ) are squares over F ( ε ), but for any g n over F [ ε ], x + εg n is not a square of apolynomial over F ( ε ), which can be seen as follows. For the sake of contradiction, assumethat x + εg n = f for some polynomial f over F ( ε ). Without loss of generality, we canassume that f is univariate, i.e., f = P di =0 c i x i , because otherwise we could set all othervariables to 0, which is a ring homomorphism. We denote by g [ i ], the coefficient of x i in anypolynomial g . Clearly, we have c = ε · g n [0]. Suppose c = 0. Therefore, c is an element ofdegree greater than 0 in F [ ε ]. Now, we consider the coefficient of x d on both sides. Clearly:2 c d c + X
So, c d c is a rational function of the form p ( ε ) q ( ε ) where both p and q are in F [ ε ] and q doesnot divide p . Thus, c d = p ( ε ) r ( ε ) where r ( ε ) in F [ ε ] has degree greater than 0 and r does notdivide p . Now consider the coefficient of x d on both sides. We get: p ( ε ) r ( ε ) = f [2 d ] = ε · g n [2 d ]But g n [2 d ] is in F [ ε ]. This gives us a contradiction! So, c = 0. But then, f = x · h where h is another polynomial over F ( ε ). Then, we have: x + εg n = x h which is clearly not possible since the left-hand side has a linear term. This completes theproof. (cid:74) We now construct ( P ) ∈ VNPC( (cid:69) p ) \ VNPC( ≤ p ) via P n := z ( y HC n + y HC n )where y and z are variables outside the set of variables in HC n , for all n . (cid:73) Lemma 7. ( P ) VNPC( ≤ p ) Proof.
This is a direct consequence of Proposition 3. (cid:74)(cid:73)
Lemma 8. ( P ) ∈ VNPC( (cid:69) p ) . Proof.
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