11 research outputs found
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
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
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
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
Multilinear Formulas, Maximal-Partition Discrepancy and Mixed-Sources Extractors
We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients ’ vector of a polynomial and the coefficients ’ vector of any product of two polynomials with disjoint sets of variables. We prove lower bounds for several old and new subclasses of circuits. Monotone Circuits: We prove a tight 2 Ω(n) lower bound for the size of monotone arithmetic circuits. The highest previous lower bound was 2 Ω( √ n). Orthogonal Formulas: We prove a tight 2 Ω(n) lower bound for the size of orthogonal multilinear formulas (defined, motivated, and studied by Aaronson). Previously, nontrivial lower bounds were only known for subclasses of orthogonal multilinear formulas. Non-Cancelling Formulas: We define and study the new model of non-cancelling multilinear formulas. Roughly speaking, in this model one is not allowed to sum two polynomials that almost cancel each other. The non-cancelling multilinear model is a generalization of both the monotone model and the orthogonal model. We prove lower bounds of n Ω(1) for the depth of non-cancelling multilinear formulas
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