How to Use Deep Learning to Identify Sufficient Conditions: A Case Study on Stanley's $e$-Positivity

TL;DR

This paper applies machine learning to identify sufficient conditions for $e$-positivity in graphs, rediscovering known conditions and proposing new conjectures, advancing algebraic combinatorics research.

Contribution

It develops a method using AI to find sufficient conditions in pure mathematics and demonstrates its effectiveness on Stanley's $e$-positivity problem.

Findings

AI identified co-triangle-free as a sufficient condition for $e$-positivity.

Saliency Map analysis suggests continuous invariants are more relevant than discrete ones.

Confirmed $e$-positivity for certain claw-free and claw-contractible-free graphs with 10 and 11 vertices.

Abstract

In a study, published in Nature, researchers from DeepMind and mathematicians demonstrated a general framework using machine learning to make conjectures in pure mathematics. Here, we build upon this framework to develop a method for identifying sufficient conditions that imply a given mathematical statement. As a demonstration, we apply this process to Stanley's problem of $e$-positivity of graphs, one of the problems that has been at the center of algebraic combinatorics for the past three decades. Guided by AI, we rediscover that one sufficient condition for a graph to be $e$-positive is that it is co-triangle-free. Based on Saliency Map analysis, we suggest that the classification of $e$-positive graphs is more related to continuous graph invariants rather than the discrete ones, which we support it with three conjectures. Furthermore, we show that the claw-free and claw-contractible-free graphs with 10 and 11 vertices are $e$-positive, resolving a conjecture by Dahlberg, Foley, and van Willigenburg.