13,964 research outputs found
Homogeneous Algebraic Complexity Theory and Algebraic Formulas
We study algebraic complexity classes and their complete polynomials under
\emph{homogeneous linear} projections, not just under the usual affine linear
projections that were originally introduced by Valiant in 1979. These
reductions are weaker yet more natural from a geometric complexity theory (GCT)
standpoint, because the corresponding orbit closure formulations do not require
the padding of polynomials. We give the \emph{first} complete polynomials for
VF, the class of sequences of polynomials that admit small algebraic formulas,
under homogeneous linear projections: The sum of the entries of the
non-commutative elementary symmetric polynomial in 3 by 3 matrices of
homogeneous linear forms.
Even simpler variants of the elementary symmetric polynomial are hard for the
topological closure of a large subclass of VF: the sum of the entries of the
non-commutative elementary symmetric polynomial in 2 by 2 matrices of
homogeneous linear forms, and homogeneous variants of the continuant polynomial
(Bringmann, Ikenmeyer, Zuiddam, JACM '18). This requires a careful study of
circuits with arity-3 product gates.Comment: This is edited part of preprint arXiv:2211.0705
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
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Projection methods in conic optimization
There exist efficient algorithms to project a point onto the intersection of
a convex cone and an affine subspace. Those conic projections are in turn the
work-horse of a range of algorithms in conic optimization, having a variety of
applications in science, finance and engineering. This chapter reviews some of
these algorithms, emphasizing the so-called regularization algorithms for
linear conic optimization, and applications in polynomial optimization. This is
a presentation of the material of several recent research articles; we aim here
at clarifying the ideas, presenting them in a general framework, and pointing
out important techniques
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