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A Note on Small Percolating Sets on Hypercubes via Generative AI

Gergely Bérczi, Adam Zsolt Wagner

TL;DR

The paper studies $r$-neighbor bootstrap percolation on the $d$-dimensional hypercube $Q_d$ and aims to tighten upper bounds on the minimum size $m(Q_d,r)$ of percolating sets. It presents a mathematical theorem showing improved upper bounds under an exact-cover condition on $[d]$, and introduces PatternBoost, a transformer-based approach, to construct smaller percolating sets for $r=4$ and $d\le13$ via data generation, ML generation, and local search refinement. The ML-driven constructions achieve percolating sets of size as small as 122 for $(d,r)=(13,4)$, with detailed observations on percolation dynamics and independence of the sets. Taken together, the work both sharpens known upper bounds for large $d$ and demonstrates a practical AI-assisted route to discover near-optimal combinatorial configurations with potential implications for related bootstrap percolation problems.

Abstract

We apply a generative AI pattern-recognition technique called PatternBoost to study bootstrap percolation on hypercubes. With this, we slightly improve the best existing upper bound for the size of percolating subsets of the hypercube.

A Note on Small Percolating Sets on Hypercubes via Generative AI

TL;DR

The paper studies -neighbor bootstrap percolation on the -dimensional hypercube and aims to tighten upper bounds on the minimum size of percolating sets. It presents a mathematical theorem showing improved upper bounds under an exact-cover condition on , and introduces PatternBoost, a transformer-based approach, to construct smaller percolating sets for and via data generation, ML generation, and local search refinement. The ML-driven constructions achieve percolating sets of size as small as 122 for , with detailed observations on percolation dynamics and independence of the sets. Taken together, the work both sharpens known upper bounds for large and demonstrates a practical AI-assisted route to discover near-optimal combinatorial configurations with potential implications for related bootstrap percolation problems.

Abstract

We apply a generative AI pattern-recognition technique called PatternBoost to study bootstrap percolation on hypercubes. With this, we slightly improve the best existing upper bound for the size of percolating subsets of the hypercube.

Paper Structure

This paper contains 5 sections, 2 theorems, 9 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The main results
  3. A mathematical theorem
  4. New constructions using machine learning
  5. Implementation of the ML pipeline for $d=13,r=4$

Key Result

Theorem 1.2

For $d\geq r \geq 1$, and hence Conjecture jozsiresult is true.

Figures (3)

  • Figure 1: A percolating set of size 122 for $d=13,r=4$. Percolation process took 68 steps
  • Figure 2: Percolation speed shows an exponential growth
  • Figure 3: Percolation process of a 122 element subset in 68 steps. Percolated nodes are colored by red. Percolation speed is exponentially increasing.

Theorems & Definitions (4)

  • Conjecture 1.1: Balogh-Bollobás Balogh2006BootstrapHypercube
  • Theorem 1.2: Morrison-Noel morrison2018extremal
  • Proposition 2.1
  • proof