A Superpolynomial Lower Bound on the Size of Uniform Non-constant-depth Threshold Circuits for the Permanent

We show that the permanent cannot be computed by DLOGTIME-uniform threshold or arithmetic circuits of depth o(log log n) and polynomial size.Comment: 11 page

Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity

We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-$3$ and depth-$4$ arithmetic circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-$3$ arithmetic circuits for a polynomial in $VNP$. 2. Exponential lower bounds for homogeneous depth-$4$ arithmetic circuits with bounded individual degree for a polynomial in $VNP$. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-$4$ arithmetic circuits for the Permanent imply a separation between ${\#}P$ and $ACC$. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean circuit complexity. Besides, it is known from a recent result of Kumar and Saptharishi [KS15] that over constant sized finite fields, strong enough average case functional lower bounds for homogeneous depth-$4$ circuits imply superpolynomial lower bounds for homogeneous depth-$5$ circuits. Our proofs are based on a family of new complexity measures called shifted evaluation dimension, and might be of independent interest

Arithmetic circuits: the chasm at depth four gets wider

In their paper on the "chasm at depth four", Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of n*n matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O(sqrt(n)*log(n)). Our depth four circuits use integer constants of polynomial size. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also give an application to boolean circuit complexity, and a simple (but suboptimal) reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree

Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity $n^{O(\log n)}$. In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of set-multilinear depth-$3$ circuits. The prior results for the sum of constantly many set-multilinear depth-$3$ circuits were only slightly better than brute-force, i.e. exponential-time. Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).Comment: 22 pages, Computational Complexity Conference, 201

Multi-k-ic depth three circuit lower bound

Abstract In a multi-k-ic depth three circuit every variable appears in at most k of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables (as the formal degree of a multi-k-ic depth three circuit can be kn where n is the number of variables). The problem of proving lower bounds for depth three circuits with high formal degree has gained in importance following a work by Gupta, Kamath, Kayal and Saptharishi [GKKS13a] on depth reduction to high formal degree depth three circuits. In this work, we show an exponential lower bound for multi-k-ic depth three circuits for any arbitrary constant k

Quasi-polynomial Hitting-set for Set-depth-Delta Formulas

We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition (X_1,...,X_d) of the variable indices [n] that the top product layer respects, i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in C respects a (unknown) partition on the variables; if Delta is even then the product gates of the bottom-most Pi-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-Delta formulas (over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta - 1}), where s is the size bound on the input set-depth-Delta formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of Delta=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth epsilon.log s / loglog s, for a fixed constant epsilon < 1. The technique is to investigate depth-Delta formulas via depth-(Delta-1) formulas over a Hadamard algebra, after applying a `shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-Delta formulas.Comment: 22 page

Unbalancing Sets and An Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

We prove a lower bound of Ω(n²/log²n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x₁,...,x_n). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([34]), who proved a lower bound of Ω(n^(4/3)/log²n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory. A special case of our combinatorial result implies, for every n, a tight Ω(n) lower bound on the minimum size of a family F of subsets of cardinality 2n of a set X of size 4n, so that any subset of X of size 2n has intersection of size exactly n with some member of F. This settles a problem of Galvin up to a constant factor, extending results of Frankl and Rödl [15] and Enomoto et al. [12], who proved in 1987 the above statement (with a tight constant) for odd values of n, leaving the even case open