Distance Metric Ensemble Learning and the Andrews-Curtis Conjecture

TL;DR

This paper introduces a novel ensemble machine learning approach to improve search strategies for the Andrews-Curtis conjecture, successfully eliminating 19 potential counterexamples and advancing understanding of this longstanding mathematical problem.

Contribution

It develops new problem-structure-based quality measures and combines them through ensemble learning to enhance search effectiveness in the Andrews-Curtis conjecture.

Findings

Eliminated 19 potential counterexamples.

Developed new heuristic measures from problem structure.

Demonstrated improved search efficiency.

Abstract

Motivated by the search for a counterexample to the Poincar\'e conjecture in three and four dimensions, the Andrews-Curtis conjecture was proposed in 1965. It is now generally suspected that the Andrews-Curtis conjecture is false, but small potential counterexamples are not so numerous, and previous work has attempted to eliminate some via combinatorial search. Progress has however been limited, with the most successful approach (breadth-first-search using secondary storage) being neither scalable nor heuristically-informed. A previous empirical analysis of problem structure examined several heuristic measures of search progress and determined that none of them provided any useful guidance for search. In this article, we induce new quality measures directly from the problem structure and combine them to produce a more effective search driver via ensemble machine learning. By this means, we eliminate 19 potential counterexamples, the status of which had been unknown for some years.