Two decades of algorithmic Feynman integral reduction

TL;DR

This historiographical review surveys two decades of algorithmic strategies for solving integration-by-parts (IBP) relations in Feynman integrals, focusing on general-purpose methods applicable to arbitrary integral families. It documents the journey from the original Laporta-based AIR/FIRE/Reduze families toward modern modular-arithmetic engines (FIRE6, FIRE7, Kira variants) that leverage distributed computing, finite-field reconstruction, and cross- CAS integration ($d=4-2\varepsilon$; $I_{a_1,\ldots,a_n} = \int\ldots\int \frac{1}{P_1^{a_1}\dots P_n^{a_n}} \prod_{i=1}^h d^d k_i$). The paper emphasizes concepts such as seed-based reductions, sector decomposition and masking strategies, and the integration of multiple backends (Fermat, Flint, Symbolica) to handle multi-loop, multi-scale calculations. The article highlights ongoing advances such as syzygy-inspired relation generation, finite-field seeding, and automated pre-solve strategies, underscoring their practical impact for scalable, high-precision perturbative quantum field theory computations.

Abstract

We present a historiographical review of algorithms and computer codes developed for solving integration-by-parts relations for Feynman integrals. This procedure is one of the key steps in the evaluation of Feynman integrals, since it enables to express integrals belonging to a given family as linear combinations of master integrals. In this review, we restrict ourselves to considering general algorithms which can, in principle, be applied to any family of Feynman integrals.