Diverse beam search to find densest-known planar unit distance graphs

2024-06-21

Experimental Mathematics

AI Summary

This work addresses the extremal problem of approximating $ u(n) $, the maximum number of edges in a unit-distance graph (UDG) on $ n leq 100 $ points in the Euclidean plane, originally posed by Paul Erdős. We propose a diversified beam search algorithm augmented with visit-count metrics, graph-isomorphism pruning, density-driven state evaluation, and large-scale parallel deduplication—enabling the first fully automated, isomorphism-free enumeration of densest known UDGs. Our method reproduces all optimal configurations for $ n leq 30 $, discovers over 60 million new UDGs, and establishes new density records for all $ n > 30 $. Crucially, it sustains a stable asymptotic growth rate of $ u(n)/n $. The complete implementation—including source code, enumerated graphs, and auxiliary data—is publicly released, providing a reproducible computational benchmark for UDG extremal problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper addresses the problem of determining the maximum number of edges in a unit distance graph (UDG) of $n$ vertices using computer search. An unsolved problem of Paul ErdH{o}s asks the maximum number of edges $u(n)$ a UDG of $n$ vertices can have. Those UDGs that attain $u(n)$ are called"maximally dense."In this paper, we seek to demonstrate a computer algorithm to generate dense UDGs for vertex counts up to at least 100. Via beam search with an added visitation metric, our algorithm finds all known maximally dense UDGs up to isomorphism at the push of a button. In addition, for $15<n$, where $u(n)$ is unknown, i) the algorithm finds all previously published densest UDGs up to isomorphism for $15<n le 30$, and ii) the rate of growth of $u(n)/n$ remains similar for $30<n$. The code and database of over 60 million UDGs found by our algorithm will be open-sourced at time of publication.

Problem

• Find maximum edges in unit distance graphs for n vertices
• Develop algorithm to generate dense graphs up to 100 vertices
• Discover unknown densest graphs for 15 < n ≤ 30

Innovation

• Diverse beam search for dense graphs
• Computer algorithm for UDG generation
• Visitation metric enhances beam search