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Finding Increasingly Large Extremal Graphs with AlphaZero and Tabu Search

Abbas Mehrabian, Ankit Anand, Hyunjik Kim, Nicolas Sonnerat, Matej Balog, Gheorghe Comanici, Tudor Berariu, Andrew Lee, Anian Ruoss, Anna Bulanova, Daniel Toyama, Sam Blackwell, Bernardino Romera Paredes, Petar Veličković, Laurent Orseau, Joonkyung Lee, Anurag Murty Naredla, Doina Precup, Adam Zsolt Wagner

TL;DR

This work tackles the extremal graph problem of maximizing edges in $n$-node graphs without 3- or 4-cycles, formalized as $f(n)=ex(n,{C3,C4})$ and optimized via a telescoping score $s(G)=e(G)-t(G)-q(G)$. It introduces an edge-flipping RL environment and a novel Pairformer architecture to guide AlphaZero, and compares it with incremental tabu search under a curriculum that seeds larger problems from smaller, high-quality graphs. Across sizes $64$ to $190$, incremental tabu search and Incremental AlphaZero push new lower bounds, often matching or surpassing the state of the art, while Wagner’s cross-entropy underperforms. The study finds curriculum-driven incremental search crucial for large state spaces and discusses the implications for learning-to-search in hard combinatorial problems, with potential extensions to other graph-design and optimization challenges.

Abstract

This work studies a central extremal graph theory problem inspired by a 1975 conjecture of Erdős, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum -- jump-starting the search for larger graphs using good graphs found at smaller sizes -- we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.

Finding Increasingly Large Extremal Graphs with AlphaZero and Tabu Search

TL;DR

This work tackles the extremal graph problem of maximizing edges in -node graphs without 3- or 4-cycles, formalized as and optimized via a telescoping score . It introduces an edge-flipping RL environment and a novel Pairformer architecture to guide AlphaZero, and compares it with incremental tabu search under a curriculum that seeds larger problems from smaller, high-quality graphs. Across sizes to , incremental tabu search and Incremental AlphaZero push new lower bounds, often matching or surpassing the state of the art, while Wagner’s cross-entropy underperforms. The study finds curriculum-driven incremental search crucial for large state spaces and discusses the implications for learning-to-search in hard combinatorial problems, with potential extensions to other graph-design and optimization challenges.

Abstract

This work studies a central extremal graph theory problem inspired by a 1975 conjecture of Erdős, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum -- jump-starting the search for larger graphs using good graphs found at smaller sizes -- we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.
Paper Structure (22 sections, 1 theorem, 3 equations, 4 figures, 4 tables, 2 algorithms)

This paper contains 22 sections, 1 theorem, 3 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. A novel neural network architecture.
  3. Incremental learning.
  4. Incremental local search.
  5. Summary of contributions.
  6. Problem Description
  7. Graph Generation as an RL Environment
  8. AlphaZero for Graph Generation
  9. Network Representation
  10. Distributed Implementation and Joint Learning Across Multiple Sizes
  11. Incremental AlphaZero
  12. Tabu Search for Graph Generation
  13. Incremental Tabu Search
  14. Experiments and Results
  15. Comparison with the state-of-the-art lower bounds.
  16. ...and 7 more sections

Key Result

Lemma 1

For any $n$-node graph $G$, we have $\operatorname{s}(G) \leq f(n)$; and there exists at least one $n$-node feasible graph for which equality holds.

Figures (4)

  • Figure 1: Normalized scores, given by $\frac{\text{number of edges}}{n\sqrt n}$, are plotted versus size, $n$. AlphaZero with curriculum (not plotted) achieves the same score as incremental tabu search for 41 of the sizes from 54 to 100. Erdős conjectured that both the red and blue curves converge to the cyan horizontal line as $n\to\infty$.
  • Figure 2: Left: Incremental tabu search, which uses a curriculum, performs increasingly better than tabu search without curriculum, for larger problem sizes. Right: Adding a curriculum improves the performance of AlphaZero significantly, especially on larger sizes.
  • Figure 3: Left: The policy cross-entropy loss of Pairformer and ResNet during online training of AlphaZero on joint training for graph sizes [80, 100] with curriculum. Pairformer minimizes the loss faster as it captures invariances and other graph structures. Right: Average episode return of Pairformer and ResNet during training of AlphaZero using the edge-flipping environment on joint training for graph sizes [80, 100] with curriculum. In both plots, the average is taken over 3 seeds and Gaussian smoothing with $\sigma=2$ is applied.
  • Figure 4: Incremental tabu search versus the cross-entropy method.

Theorems & Definitions (2)

  • Lemma 1
  • Definition 1: $\bigoplus$, flipping