Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.01257

A Numerical Refinement of an Exact Discrete Fourier Transform Bound

Zhihao Yao·Sichao Liu
View original source ↗Read source document ↗Export metadata ↓

Abstract

We refine the numerical bounds in Kartal's instantiation of Sussman's exact discrete Fourier transform construction. For every integer n >= 1, the resulting algorithm computes the length-n transform with work O(n (log n)^(1 - delta)), where delta = 0.0030201659. The bound holds in an exact complex-arithmetic RAM model with a supplied canonical root of unity. This final exponent saving exceeds the earlier certified parameter C_0 = 0.00302016 by 5.9 x 10^(-9). We retain the underlying circuit, tighten rational bounds for sums of powers of its block and target ranks, and increase the fixed tensor power from 7760 to 11640. These bounds establish the inequalities for Sussman's recursion at the intermediate certificate parameter C = 0.00302016595. The gap C - delta absorbs the overhead of recursion and Fourier reduction. The resulting transform theorem is proved in Lean and checked against the reference problem statement.

Explore related subjects

cs.CCcs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhihao Yao, Sichao Liu. 2026-10-31. A Numerical Refinement of an Exact Discrete Fourier Transform Bound. https://arxiv.org/abs/2611.01257

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

NP-Hardness and Additive Inapproximability of Deterministic Communication Complexity

Given the full truth table of a Boolean function whose input is split between two players, we study the complexity of determining the minimum worst-case number of bits they must communicate to compute it. This is a basic meta-complexity question, posed by Yao in 1979. We prove NP-hardness of approximation within every fixed additive error for deterministic communication complexity, under deterministic polynomial-time reductions. The result holds both in the standard model and in Yao's original model, where the players alternate after every bit and only the receiver of the last bit must determine the answer. In particular, it resolves the exact computation question under Yao's original conventions. Earlier versions of this same paper established exact NP-hardness for the standard model, independently and concurrently with Hirahara, Ilango, and Loff. In both models, a polynomial-time estimator with any fixed additive error would yield a polynomial-time $3$-SAT algorithm, deterministic or bounded-error randomized according to the estimator. For the standard model, we also give quantitative implications for errors that grow with the number $T$ of table entries. A bounded-error randomized polynomial-time estimator with additive error $c\log T/\log\log T$, for any fixed $0\le c<1/8$, yields a bounded-error randomized $3$-SAT algorithm on $N$ variables and $O(N)$ clauses running in time $2^{O(N^α)}$ for every fixed $8c<α<1$. A deterministic polynomial-time estimator with additive error $c_*\log\log T$, for an effective absolute constant $c_*>0$, yields a deterministic algorithm running in time $2^{O((\log N)^2)}$ on such formulas.

cs.CC↗
arXiv2026

On the average-case complexity landscape for Tensor-Isomorphism-complete problems over finite fields

In Grochow and Qiao (\emph{SIAM J. Comput.}, 2023), the complexity class Tensor Isomorphism (TI) was introduced and isomorphism problems for groups, algebras, and polynomials were shown to be TI-complete. In this paper, we study average-case algorithms for several TI-complete problems over finite fields, including algebra isomorphism, matrix code conjugacy, and $4$-tensor isomorphism. Our main results reveal a rich landscape of average-case complexities of Tensor Isomorphism problems, despite their equivalence in the worst-case setting. Over the finite field of order $q$, we devise (1) average-case \emph{polynomial-time} algorithms for algebra isomorphism and matrix code conjugacy that succeed for $1/Θ(q)$ fraction of inputs; (2) an average-case \emph{polynomial-time} algorithm for the $4$-tensor isomorphism that succeeds for $1/q^{Θ(1)}$ fraction of inputs; and (3) an average-case \emph{polynomial-time} algorithm for the orthogonal $3$-tensor isomorphism that succeed for $1-1/\poly(n)$ fraction of $n\times n\times n$ tensors. Our main technical contribution is to introduce the spectral properties of random matrices into algorithms for TI-complete problems. This leads to not only new algorithms but also new questions in random matrix theory over finite fields. To settle these questions, we need to extend both the generating function approach as in Neumann and Praeger (\emph{J. London Math. Soc.}, 1998) and the character sum method of Gorodetsky and Rodgers (\emph{Trans. Amer. Math. Soc.}, 2021).

cs.CC↗
arXiv2026

Constant-factor approximation of MinCostCSP with a conservative majority polymorphism

For a relational structure A, the Minimum Cost Constraint Satisfaction Problem is the following problem denoted by MinCostCSP(A): Given an instance of CSP(A) with rational costs on variable-value pairs, find a solution to the instance minimizing the sum of the chosen costs. For the exact minimization, a classification of MinCostCSP(A) in terms of A was established by Takhanov [STACS'10]. We focus on constant-factor approximations of MinCostCSP(A). DeHaan, Huang, and Lee recently showed that if A fails to admit a conservative near-unanimity polymorphism then MinCostCSP(A) is not constant-factor approximable [APPROX'25]. We provide a significant step towards a full classification, by proving a dichotomy for structures A admitting a conservative majority (also known as 3-near-unanimity) polymorphism. Our dichotomy criterion is not in terms of an algebraic condition on A but we show that this is unavoidable. We include a simple argument proving that no such condition exists.

cs.CC↗