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Constructions in combinatorics via neural networks

Adam Zsolt Wagner

TL;DR

The paper demonstrates that reinforcement learning, via the deep cross-entropy method, can yield explicit constructions and counterexamples in extremal combinatorics and graph theory without problem-specific priors. It encodes problems as word-generation tasks, compares cross-entropy with other RL methods, and applies it to multiple conjectures, producing concrete counterexamples and structural insights. In addition to neural methods, it showcases LP solver-driven constructions for linear-programmable questions. The work highlights the potential of AI-assisted constructive mathematics and suggests avenues for applying alternative RL algorithms to open mathematical conjectures.

Abstract

We demonstrate how by using a reinforcement learning algorithm, the deep cross-entropy method, one can find explicit constructions and counterexamples to several open conjectures in extremal combinatorics and graph theory. Amongst the conjectures we refute are a question of Brualdi and Cao about maximizing permanents of pattern avoiding matrices, and several problems related to the adjacency and distance eigenvalues of graphs.

Constructions in combinatorics via neural networks

TL;DR

The paper demonstrates that reinforcement learning, via the deep cross-entropy method, can yield explicit constructions and counterexamples in extremal combinatorics and graph theory without problem-specific priors. It encodes problems as word-generation tasks, compares cross-entropy with other RL methods, and applies it to multiple conjectures, producing concrete counterexamples and structural insights. In addition to neural methods, it showcases LP solver-driven constructions for linear-programmable questions. The work highlights the potential of AI-assisted constructive mathematics and suggests avenues for applying alternative RL algorithms to open mathematical conjectures.

Abstract

We demonstrate how by using a reinforcement learning algorithm, the deep cross-entropy method, one can find explicit constructions and counterexamples to several open conjectures in extremal combinatorics and graph theory. Amongst the conjectures we refute are a question of Brualdi and Cao about maximizing permanents of pattern avoiding matrices, and several problems related to the adjacency and distance eigenvalues of graphs.

Paper Structure

This paper contains 13 sections, 4 theorems, 23 equations, 13 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Reinforcement learning and the cross-entropy method
  3. Applying the cross-entropy method to problems in extremal combinatorics
  4. Example: the sum of matching number and largest eigenvalue of graphs
  5. A conjecture about the proximity and distance eigenvalues of graphs
  6. The peaks of distance and adjacency polynomials of trees can be far apart
  7. Transmission regularity and the distance Laplacian
  8. The permanent of 312-pattern avoiding 0-1 matrices
  9. Finding constructions with LP solvers
  10. An exact covering problem in the hypercube
  11. Weakly cross-intersecting (a,b)-set systems
  12. Concluding remarks
  13. The construction for Problem \ref{['prob:domotor']}

Key Result

Theorem 2.2

If $T$ is a tree on $n$ vertices with distance matrix $D(T)$, then

Figures (13)

  • Figure 1: We are generating a graph, and so far the network has included edges 1,2,3,6, and rejected edges 4,5 (dotted), corresponding to the sequence 111001. We input this into the neural network to get a probability distribution on whether to include or reject the next edge.
  • Figure 2: The average value of $\lambda_1+\mu$ of the best 10% of the sessions in each iteration decreases over time. After 5000 iterations it goes slightly below the conjectured threshold of $\sqrt{19-1}+1$, meaning we have found a counterexample to Conjecture \ref{['conj:aouch']}.
  • Figure 3: The evolution of the best construction over time. The network quickly realizes that sparse graphs are best, and eventually the "balanced double star" structure emerges.
  • Figure 4: A graph on 19 vertices satisfying $\lambda_1 + \mu < \sqrt{n-1}+1$.
  • Figure 5: The graph on 30 vertices with smallest value of $\pi + \partial_{\mathopen{}\mathclose{\left\lfloor \frac{2D}{3} \right\rfloor}}$ found by the network. It has $\pi + \partial_{\mathopen{}\mathclose{\left\lfloor \frac{2D}{3} \right\rfloor}}\approx 0.4$ so it is not quite a counterexample to Conjecture \ref{['conj:aouch2']}, but it tells us very clearly what counterexamples could look like.
  • ...and 8 more figures

Theorems & Definitions (12)

  • Conjecture 2.1: aouch
  • Theorem 2.2: Graham-Pollack grahampollack
  • Conjecture 2.3: Auchiche--Hansen aouchhansen
  • Conjecture 2.4: Collins collins
  • Theorem 2.5
  • Proposition 2.6
  • proof
  • proof : Proof of Theorem \ref{['thm:collins']}
  • Conjecture 2.7: Shor collins
  • Proposition 2.9
  • ...and 2 more