Towards a tight hardness–randomness connection between permanent and arithmetic circuit identity testing

Maurice Jansen
2012 Information Processing Letters  
In this paper we make progress on establishing a tight connection between the problem of derandomization of arithmetic circuit identity testing (ACIT), and the arithmetic circuit complexity of the permanent defined by per n = σ∈Sn n i=1 x iσ(i) . We develop an ACIT-based derandomization hypothesis, and show this is a necessary condition for proving that permanent has super-polynomial arithmetic circuits over F, for fields F of characteristic zero. Informally, this hypothesis poses the existence
more » ... of a subexponential size hitting set 1 H n computable by subexponential size uniform TC 0 circuits against size n arithmetic circuits with m ≤ n variables whose output is multilinear. Assuming the Generalized Riemann Hypothesis (GRH), it can be shown that this hypothesis is sufficient for showing that either permanent does not have polynomial size (nonuniform) arithmetic circuits, or that the Boolean circuit class uniform TC 0 is strictly contained in uniform NC 2 . Without (GRH), the hypothesis implies such a disjunction, but with the first item stating permanent does not have polynomial size constant-free 2 arithmetic circuits. In this setting the converse also goes through, but based on the slightly stronger assumption that all constant multiples a n · per n require super-polynomial constant-free arithmetic circuits, for a n ∈ Z/{0} computable by poly(n) size constant-free circuits. the problem of proving super-polynomial arithmetic circuit size lower bounds for the permanent polynomial per n = σ∈Sn n i=1 x iσ(i) . Permanent is complete for Valiant's algebraic complexity class VNP F , which is the nondeterminstic counterpart of the class VP F of poly degree polynomials computable by poly size arithmetic circuits over F. Proving VP F = VNP F is the central open problem in algebraic complexity theory, and this is equivalent to showing that L F (per n ) = n O(1) , cf. [BCS97]. Arithmetic circuit identity testing (ACIT) over F is the problem of deciding for a given arithmetic circuit Φ over F, whether the polynomial computed by Φ is identical to the zero polynomial of F[X]. This problem is efficiently solvable using randomization. Using the Schwartz-Zippel-deMillo-Lipton Lemma [DL78, Sch80, Zip79], Ibarra and Moran [IM83] show the problem is in coRP. Derandomization of ACIT and the problem of proving explicit circuit lower bounds are closely connected. This connection already appears in 1980s work of Heintz and Schnorr [HS80]. In recent years our understanding of this connection has improved remarkably. In a landmark paper, Kabanets and Impagliazzo [KI04] show that giving an NSUBEXP time algorithm for ACIT(Q) implies that either the permanent polynomial requires super-polynomial arithmetic circuits over Q, or that NEXP ⊆ P/poly. Agrawal [Agr05] shows that explicitly constructing a poly(n) size hitting set against the class of size n arithmetic circuits yields an exponential arithmetic circuit size lower bound for a polynomial with coefficients computable in PSPACE. Going in the converse direction, Ref. [KI04] shows how arithmetic lower bounds can be leveraged to yield low-degree identity testing by using Nisan-Wigderson designs [NW94] and a result about polynomial factorization independently obtained by Kaltofen [Kal89] and Bürgisser [Bür04]. The above mentioned results make it conceivable that the statement L F (per n ) = n O(1) is equivalent to some ACIT-based derandomization assumption. In this paper we make progress towards this aim. We establish a necessary derandomization condition for proving super-polynomial lower bounds for permanent. For the converse, we prove a result similar in flavor to Ref. [KI04], but for the black-box 3 setting. Namely, we show that our derandomization condition is sufficient to imply that either permanent requires super-polynomial (nonuniform) arithmetic circuit, or that the Boolean circuit class uniform TC 0 is strictly contained in uniform NC 2 .
doi:10.1016/j.ipl.2012.08.001 fatcat:iyw4sp5iarbgza6xo7g4gkejoi