RLGT: A reinforcement learning framework for extremal graph theory

TL;DR

RLGT is a new reinforcement learning framework designed to advance research in extremal graph theory by supporting diverse graph types and improving computational efficiency.

Contribution

It systematizes previous RL approaches in graph theory, offering a flexible, efficient, and modular framework for future research in the field.

Findings

Refuted inequalities concerning the Laplacian spectral radius of graphs.

Obtained new lower bounds for certain Ramsey numbers.

Contributed to Turán-type extremal problems involving cycles.

Abstract

Reinforcement learning (RL) is a subfield of machine learning that focuses on developing models that can autonomously learn optimal decision-making strategies over time. In a recent pioneering paper, Wagner demonstrated how the Deep Cross-Entropy RL method can be applied to tackle various problems from extremal graph theory by reformulating them as combinatorial optimization problems. Subsequently, many researchers became interested in refining and extending the framework introduced by Wagner, thereby creating various RL environments specialized for graph theory. Moreover, a number of problems from extremal graph theory were solved through the use of RL. In particular, several inequalities concerning the Laplacian spectral radius of graphs were refuted, new lower bounds were obtained for certain Ramsey numbers, and contributions were made to the Tur\'{a}n-type extremal problem in which the forbidden structures are cycles of length three and four. Here, we present Reinforcement Learning for Graph Theory (RLGT), a novel RL framework that systematizes the previous work and provides support for both undirected and directed graphs, with or without loops, and with an arbitrary number of edge colors. The framework efficiently represents graphs and aims to facilitate future RL-based research in extremal graph theory through optimized computational performance and a clean and modular design.