Turing in the shadows of Nobel and Abel: an algorithmic story behind two recent prizes

TL;DR

This paper explores how the groundbreaking work of Nobel and Abel laureates Parisi and Talagrand has profoundly influenced the understanding of algorithms for optimization problems involving randomness, revealing which problems are efficiently solvable.

Contribution

It highlights the algorithmic implications of Parisi and Talagrand's work, providing a precise characterization of problems with fast algorithms versus those without.

Findings

Revolutionized understanding of optimization algorithms involving randomness

Provided criteria for the existence of fast algorithms for certain problems

Clarified why some problems resist efficient algorithmic solutions

Abstract

The 2021 Nobel Prize in physics was awarded to Giorgio Parisi ``for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales,'' and the 2024 Abel Prize in mathematics was awarded to Michel Talagrand ``for his groundbreaking contributions to probability theory and functional analysis, with outstanding applications in mathematical physics and statistics.'' What remains largely absent in the popular descriptions of these prizes, however, is the profound contributions the works of both individuals have had to the field of \emph{algorithms and computation}. The ideas first developed by Parisi and his collaborators relying on remarkably precise physics intuition, and later confirmed by Talagrand and others by no less remarkable mathematical techniques, have revolutionized the way we think algorithmically about optimization problems involving randomness. This is true both in terms of the existence of fast algorithms for some optimization problems, but also in terms of our persistent failures of finding such algorithms for some other optimization problems. The goal of this article is to highlight these developments and explain how the ideas pioneered by Parisi and Talagrand have led to a remarkably precise characterization of which optimization problems admit fast algorithms, versus those which do not, and furthermore to explain why this characterization holds true.