Related papers: Proof Complexity Lower Bounds from Algebraic Circu…

This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that…

The computational complexity of polynomial ideals and Gr\"obner bases has been studied since the 1980s. In recent years, the related notions of polynomial subalgebras and SAGBI bases have gained more and more attention in computational…

We present a generic way to obtain level lower bounds for (promise) CSP hierarchies from degree lower bounds for algebraic proof systems. More specifically, we show that pseudo-reduction operators in the sense of Alekhnovich and Razborov…

We study implicational formulas in the context of proof complexity of intuitionistic propositional logic (IPC). On the one hand, we give an efficient transformation of tautologies to implicational tautologies that preserves the lengths of…

In this note, we propose a framework for proving computational lower bounds in norm approximation by leveraging a reverse detection--estimation gap. The starting point is a testing problem together with an estimator whose error is…

Polynomial factorization is a fundamental problem in computational algebra. Over the past half century, a variety of algorithmic techniques have been developed to tackle different variants of this problem. In parallel, algebraic complexity…

A growing trend in program analysis is to encode verification conditions within the language of the input program. This simplifies the design of analysis tools by utilizing off-the-shelf verifiers, but makes communication with the…

We show new upper and lower bounds for the complexity of implementation of a sequence of Boolean matrices proposed by Kaski et al. (arXiv:1208.0554) with additive circuits.

In the complexity estimation for an attack that reduces a cryptosystem to solving a system of polynomial equations, the degree of regularity and an upper bound of the first fall degree are often used in cryptanalysis. While the degree of…

As state-of-the-art neural networks are deployed on reasoning and algorithmic tasks, exactness guarantees become increasingly important. However, high average-case accuracy can still mask inconsistent behaviors. This motivates exact…

This paper deals with the polynomial linear system solving with errors (PLSwE) problem. Specifically, we focus on the evaluation-interpolation technique for solving polynomial linear systems and we assume that errors can occur in the…

We prove new upper bounds for the degrees in Hilbert's Nullstellensatz and for the Noether exponent of polynomial ideals in terms of the monomial structure of the polynomials involved. Our bounds improve the previously known bounds in the…

Asymptotically tight lower bounds are derived for the I/O complexity of a general class of hybrid algorithms computing the product of $n \times n$ square matrices combining ``\emph{Strassen-like}'' fast matrix multiplication approach with…

We study arithmetic proof systems P_c(F) and P_f(F) operating with arithmetic circuits and arithmetic formulas, respectively, that prove polynomial identities over a field F. We establish a series of structural theorems about these proof…

In recent years, we have seen several approaches to the graph isomorphism problem based on "generic" mathematical programming or algebraic (Gr\"obner basis) techniques. For most of these, lower bounds have been established. In fact, it has…

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates.…

We provide accurate upper bounds on the Boolean circuit complexity of the standard and the Karatsuba methods of integer multiplication

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…

We propose an algebraic approach to proving circuit lower bounds for ACC0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC0 and ACC0 can be reformulated in…

Low-rank pseudoinverses are widely used to approximate matrix inverses in scalable machine learning, optimization, and scientific computing. However, real-world matrices are often observed with noise, arising from sampling, sketching, and…