28 research outputs found
Arithmetic Circuits and the Hadamard Product of Polynomials
Motivated by the Hadamard product of matrices we define the Hadamard product
of multivariate polynomials and study its arithmetic circuit and branching
program complexity. We also give applications and connections to polynomial
identity testing. Our main results are the following. 1. We show that
noncommutative polynomial identity testing for algebraic branching programs
over rationals is complete for the logspace counting class \ceql, and over
fields of characteristic the problem is in \ModpL/\Poly. 2.We show an
exponential lower bound for expressing the Raz-Yehudayoff polynomial as the
Hadamard product of two monotone multilinear polynomials. In contrast the
Permanent can be expressed as the Hadamard product of two monotone multilinear
formulas of quadratic size.Comment: 20 page
On Isoperimetric Profiles and Computational Complexity
The isoperimetric profile of a graph is a function that measures, for an integer k, the size of the smallest edge boundary over all sets of vertices of size k. We observe a connection between isoperimetric profiles and computational complexity. We illustrate this connection by an example from communication complexity, but our main result is in algebraic complexity.
We prove a sharp super-polynomial separation between monotone arithmetic circuits and monotone arithmetic branching programs. This shows that the classical simulation of arithmetic circuits by arithmetic branching programs by Valiant, Skyum, Berkowitz, and Rackoff (1983) cannot be improved, as long as it preserves monotonicity.
A key ingredient in the proof is an accurate analysis of the isoperimetric profile of finite full binary trees. We show that the isoperimetric profile of a full binary tree constantly fluctuates between one and almost the depth of the tree
Circuit Lower Bounds, Help Functions, and the Remote Point Problem
We investigate the power of Algebraic Branching Programs (ABPs) augmented
with help polynomials, and constant-depth Boolean circuits augmented with help
functions. We relate the problem of proving explicit lower bounds in both these
models to the Remote Point Problem (introduced by Alon, Panigrahy, and Yekhanin
(RANDOM '09)). More precisely, proving lower bounds for ABPs with help
polynomials is related to the Remote Point Problem w.r.t. the rank metric, and
for constant-depth circuits with help functions it is related to the Remote
Point Problem w.r.t. the Hamming metric. For algebraic branching programs with
help polynomials with some degree restrictions we show exponential size lower
bounds for explicit polynomials
Notes on Boolean Read-k and Multilinear Circuits
A monotone Boolean (OR,AND) circuit computing a monotone Boolean function f
is a read-k circuit if the polynomial produced (purely syntactically) by the
arithmetic (+,x) version of the circuit has the property that for every prime
implicant of f, the polynomial contains at least one monomial with the same set
of variables, each appearing with degree at most k. Every monotone circuit is a
read-k circuit for some k. We show that already read-1 (OR,AND) circuits are
not weaker than monotone arithmetic constant-free (+,x) circuits computing
multilinear polynomials, are not weaker than non-monotone multilinear
(OR,AND,NOT) circuits computing monotone Boolean functions, and have the same
power as tropical (min,+) circuits solving combinatorial minimization problems.
Finally, we show that read-2 (OR,AND) circuits can be exponentially smaller
than read-1 (OR,AND) circuits.Comment: A throughout revised version. To appear in Discrete Applied
Mathematic
Shadows of Newton Polytopes
We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of P to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity
Non-malleable coding against bit-wise and split-state tampering
Non-malleable coding, introduced by Dziembowski et al. (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable coding is possible against any class of adversaries of bounded size. In particular, Dziembowski et al. show that such codes exist and may achieve positive rates for any class of tampering functions of size at most (Formula presented.), for any constant (Formula presented.). However, this result is existential and has thus attracted a great deal of subsequent research on explicit constructions of non-malleable codes against natural classes of adversaries. In this work, we consider constructions of coding schemes against two well-studied classes of tampering functions; namely, bit-wise tampering functions (where the adversary tampers each bit of the encoding independently) and the much more general class of split-state adversaries (where two independent adversaries arbitrarily tamper each half of the encoded sequence). We obtain the following results for these models. (1) For bit-tampering adversaries, we obtain explicit and efficiently encodable and decodable non-malleable codes of length n achieving rate (Formula presented.) and error (also known as “exact security”) (Formula presented.). Alternatively, it is possible to improve the error to (Formula presented.) at the cost of making the construction Monte Carlo with success probability (Formula presented.) (while still allowing a compact description of the code). Previously, the best known construction of bit-tampering coding schemes was due to Dziembowski et al. (ICS 2010), which is a Monte Carlo construction achieving rate close to .1887. (2) We initiate the study of seedless non-malleable extractors as a natural variation of the notion of non-malleable extractors introduced by Dodis and Wichs (STOC 2009). We show that construction of non-malleable codes for the split-state model reduces to construction of non-malleable two-source extractors. We prove a general result on existence of seedless non-malleable extractors, which implies that codes obtained from our reduction can achieve rates arbitrarily close to 1 / 5 and exponentially small error. In a separate recent work, the authors show that the optimal rate in this model is 1 / 2. Currently, the best known explicit construction of split-state coding schemes is due to Aggarwal, Dodis and Lovett (ECCC TR13-081) which only achieves vanishing (polynomially small) rate