A Systematization of the Wagner Framework: Graph Theory Conjectures and Reinforcement Learning

Flora Angileri, Giulia Lombardi, Andrea Fois, Renato Faraone, Carlo Metta, Michele Salvi, Luigi Amedeo Bianchi, Marco Fantozzi, Silvia Giulia Galfrè, Daniele Pavesi, Maurizio Parton, Francesco Morandin

Abstract: In 2021, Adam Zsolt Wagner proposed an approach to disprove conjectures in graph theory using Reinforcement Learning (RL). Wagner frames a conjecture as f(G) < 0 for every graph G, for a certain invariant f; one can then play a single-player graph-building game, where at each turn the player decides whether to add an edge or not. The game ends when all edges have been considered, resulting in a certain graph \(G_T\), and \(f(G_T)\) is the final score of the game; RL is then used to maximize this score. This brilliant idea is as simple as innovative, and it lends itself to systematic generalization. Several different single-player graph-building games can be employed, along with various RL algorithms. Moreover, RL maximizes the cumulative reward, allowing for step-by-step rewards instead of a single final score, provided the final cumulative reward represents the quantity of interest \(f(G_T)\). In this paper, we discuss these and various other choices that can be significant in Wagner’s framework. As a contribution to this systematization, we present four distinct single-player graph-building games. Each game employs both a step-by-step reward system and a single final score. We also propose a principled approach to select the most suitable neural network architecture for any given conjecture and introduce a new dataset of graphs labeled with their Laplacian spectra. The games have been implemented as environments in the Gymnasium framework, and along with the dataset and a simple interface to play with the environments, are available at https://github.com/CuriosAI/graph_conjectures.