A Quadratic Lower Bound for the Permanent and Determinant Problem over any Characteristic 6= 2 Jin-Yi Cai Xi Chen Dong Li Computer Sciences School of Mathematics School of Mathematics Department, University of Institute for Advanced Study Institute for Advanced Study Wisconsin, Madison U.S.A. U.S.A. and Radcliffe Institute [email protected] [email protected] Harvard University, U.S.A. [email protected] ABSTRACT is also well-studied, especially in combinatorics [12]. For In Valiant’s theory of arithmetic complexity, the classes VP example, if A is a 0-1 matrix then per(A) counts the number and VNP are analogs of P and NP. A fundamental problem of perfect matchings in a bipartite graph with adjacency A concerning these classes is the Permanent and Determinant matrix . Problem: Given a field F of characteristic = 2, and an inte- These well-known functions took on important new mean- ger n, what is the minimum m such that the6 permanent of ings when viewed from the computational complexity per- spective. It is well known that the determinant can be com- an n n matrix X =(xij ) can be expressed as a determinant of an×m m matrix, where the entries of the determinant puted in polynomial time. In fact it can be computed in the × complexity class NC2. By contrast, Valiant [22, 21] showed matrix are affine linear functions of xij ’s, and the equal- ity is in F[X]. Mignon and Ressayre (2004) [11] proved a that computing the permanent is #P-complete. quadratic lower bound m = Ω(n2) for fields of characteristic In fact, Valiant [21] (see also [4, 5]) has developed a sub- 0. We extend the Mignon-Ressayre quadratic lower bound stantial theory. The complexity classes VPF and VNPF are to all fields of characteristic = 2. the analogs of P and NP in this theory of arithmetic com- 6 plexity, and det and per functions are the central objects in the two classes, respectively. It was shown that the com- Categories and Subject Descriptors plexity of computing the permanent characterizes the class F.2.1 [Analysis of Algorithms and Problem Complex- VNPF and the complexity of computing the determinant (al- ity]: Numerical Algorithms and Problems—Computations most) characterizes the class VPF. in finite fields, Computations on matrices, Computations on More precisely, a family of polynomials fn is in VPF if O(1) { } polynomials deg(fn)= n and there is a family of arithmetic circuits O(1) of size n computing fn . A family of polynomials gn { O}(1) { } General Terms is in VNPF if deg(gn) = n , and there exists a family of polynomials fn VPF such that Theory { } ∈ gn(x1, ..., xn)= fn+m(x1, ..., xn,y1, ..., ym), Keywords y ,...,ym∈{0,1} 1 X O(1) Permanent, determinant, arithmetic complexity, finite field where m = n . We say that fn is a projection of gm { } { } if there are some α1, α2,...,αm F x1,...,xn , such ∈ ∪ { } 1. INTRODUCTION that fn(x1,...,xn)= gm(α1,...,αm). It isa p-projection if 2 O(1) Given a set of n indeterminates X =(xi,j )i,j=1,...,n over m = n . A projection is a particularly simple reduction. a field F, we can define It is a special case of an affine linear reduction, where each αi is an affine linear function of xi’s. Valiant proved that det(X)= sign(π) n x , and π∈Sn i=1 i,π(i) n For any field F, F. per(X)= x . Theorem 1 (Valiant). per VNP P π∈Sn i=1Q i,π(i) ∈ Moreover, for any F with char F = 2, any fn VNPF is 6 { } ∈ The determinant functionP is certainlyQ one of the most well- a p-projection of per. studied functions in mathematics. The permanent function It is also known that det is in VPF [23]. More exact char- acterizations of det were given in terms of polynomial-sized arithmetic branching programs [7, 19, 24]. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are Theorem 2 (Valiant). Any polynomial fn is a pro- not made or distributed for profit or commercial advantage and that copies jection of detm of an m m matrix, where m is linear in × bear this notice and the full citation on the first page. To copy otherwise, to the formula size of fn. In particular, if fn has polyno- republish, to post on servers or to redistribute to lists, requires prior specific { } mial formula size, then fn is a p-projection of det. Also permission and/or a fee. { } STOC’08, if fn VPF, then fn is the projection of detm for some May 17–20, 2008, Victoria, British Columbia, Canada. { }O ∈(log n) Copyright 2008 ACM 978-1-60558-047-0/08/05 ...$5.00. m = n . 491 2 n By Ryser’s formula [12], pern has formula size O(n 2 ). and the matrix H becomes 0. In this paper we overcome Thus by Valiant’s theorem it is the projection of detm, where this difficulty by considering another explicit construction 2 n m = O(n 2 ). Furthermore if we view Ryser’s formula as of matrix X0. on the truncated linear row sums directly (instead of on the We mention some other related results. In [9], Jerrum and variables), then Valiant’s theorem implies that Snir showed that any monotone arithmetic circuit family that computes permanent must have exponential size. For Theorem 3. For every n, there exists a collection A of 2 depth-three arithmetic circuits over fields of characteristic 0, affine linear functions Ak,l(X) over n variables, where 1 n ≤ Shpilka and Wigderson [17] proved that the permanent (and k,l m = O(2 ), such that per (X) = detm(A(X)). 2 ≤ n determinant) requires circuit size Ω(n ). For depth-three It is remarkable that this is the best general upper bound arithmetic circuits over finite characteristic, Grigoriev and known for this. Razborov [8] showed an exponential lower bound for both determinant and permanent. Raz [15] proved a lower bound O(log n) Definition 1. The determinantal complexity dc of fn is of n on the size of families of multilinear formulas the minimum integer m, such that there exist affine linear computing permanent and determinant. For syntactically functions Ak,l(X), where 1 k,l m, such that fn(X) = multilinear arithmetic circuits, Raz, Shpilka and Yehuday- ≤ ≤ 4/3 2 detm(A(X)). off [16] proved a Ω(n / log n) lower bound for an explicit multilinear function. A survey of some work on the Perma- dc The question addressed in this paper is about (pern). nent and Determinant Problem can be found in [1], where Valiant’s analog of P = NP will follow if one can show a it also discusses an algebraic geometry approach by Mulmu- dc 6 ω(log n) lower bound (pern)= n . ley and Sohoni [13] and connections to the pseudorandom In some sense this problem has a longer history. P´olya [14] generator used in the AKS proof for primality [2, 3]. was the first to ask a question on when one can express a This paper is organized as follows. In Section 2, we discuss permanent as a modified determinant. He noticed that the general approach by Mignon and Ressayre, and state our 2 a b a b result. In Section 3, we prove an Ω(n ) lower bound valid per = det , c d c− d for all characteristic = 2. In Section 4 we indicate how to improve the leading constant6 in Ω(n2) to match the Mignon- and asked if there are any similar equations, by affixing 1 to Ressayre bound. the variables, for n 3. This was answered in the negative± ≥ by Szeg¨o [18]. This line of inquiry culminated in 2. THE APPROACH AND THE THEOREM Theorem 4 (Marcus, Minc). If char F = 0 and n ≥ 3, then there are no homogeneous linear functions fk,ℓ in the 2.1 The Proof by Mignon and Ressayre indeterminates xi,j (1 i,j,k,ℓ n) such that per(xi,j )= Given an n n matrix X = (xi,j )i,j=1,2,...,n over a field ≤ ≤ × det(fk,ℓ). F, the determinant det(X) and the permanent per(X) are 2 dc both polynomials of degree n over n variables. Their partial In terms of (pern), this celebrated theorem is equivalent to dc(per ) n + 1, over fields of char F = 0 (note that if derivatives of all orders are defined formally. n H X Hes- the permanental≥ matrix is also n n, then clearly constant We use ( )=(Hij,kl)i,j,k,l=1,2,...,n to denote the sian X terms in affine linear equations do× not help, as seen by the matrix of per( ): homogeneous part.). ∂2per(X) dc F X The first non-trivial lower bound for (pern) is by von zur Hij,kl = [ ], for all 1 i,j,k,l n. ∂xi,j ∂xk,l ∈ ≤ ≤ dc Gathen [25], who showed that (pern) 8/7n. (This was proved for p-projections.) Von zur Gathen’s≥ result was then Similarly, we can define the Hessian matrix of det(X), and p improved independently by Babai and Seress (as reported denote it by Hdet(X). in [26]), by Cai [6], and by Meshulam [10]. Their results Now suppose that there exists a collection A of m2 affine were (ignoring lower order terms) dc(per ) √2n. linear functions, where A = Ak,l(x1,1,x1,2,...,xn,n), k,l : n { This rather weak lower bound stood as≥ the best bound 1 k,l m , such that in the polynomial ring F[X], ≤ ≤ } until 2004, when Mignon and Ressayre [11] proved that 2 dc(per ) n /2, over any field of char F = 0.