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Reinforcement Learning the Chromatic Symmetric Function

Gergely Bérczi, Jonas Klüver

Abstract

We propose a conjectural counting formula for the coefficients of the chromatic symmetric function of unit interval graphs using reinforcement learning. The formula counts specific disjoint cycle-tuples in the graphs, referred to as Eschers, which satisfy certain concatenation conditions. These conditions are identified by a reinforcement learning model and are independent of the particular unit interval graph, resulting a universal counting expression.

Reinforcement Learning the Chromatic Symmetric Function

Abstract

We propose a conjectural counting formula for the coefficients of the chromatic symmetric function of unit interval graphs using reinforcement learning. The formula counts specific disjoint cycle-tuples in the graphs, referred to as Eschers, which satisfy certain concatenation conditions. These conditions are identified by a reinforcement learning model and are independent of the particular unit interval graph, resulting a universal counting expression.

Paper Structure

This paper contains 23 sections, 17 theorems, 67 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The main results
  3. Mathematical background
  4. Stanley's G-homomorphism
  5. Eschers
  6. Escher-pairs
  7. Escher-tuples
  8. Machine Learning
  9. The RL algorithm
  10. Implementation
  11. Score Function
  12. Hyperparameters
  13. Results
  14. Analysis of the Machine Learning Method
  15. Non-Uniqueness of Solutions and Balancing Complexity
  16. ...and 8 more sections

Key Result

Theorem 1.9

A finite poset $P$ is a UIO if and only if it is $2+2$-free and $3+1$-free.

Figures (4)

  • Figure 1: Insertion points and sub-Eschers for the Escher pair $u=[4,2,8,6,10,9,7], v=[0,5,3,1]$ in the UIO $U=(0,0,1,1,2,3,3,4,6,7,9)$. Green dots indicate insertion points (i.e $2 \to 3$ and $5 \to 8$ in $U$, and underbrace indicates length $4$ sub-Eschers in $u$.
  • Figure 2: Condition graph for Escher pairs
  • Figure 3: The condition graphs for Escher triples
  • Figure 4: Training charts for the partition $(5,2,1)$ trained on all UIOs of length $8$, with 3 rows in the condition graph. The number of correctly predicted coefficients is $3848/4862$. Because EdgePenalty<1, the model occasionally adds edges to the condition graph to achieve a lower total score, resulting in jumps in the number of edges.

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9: Scott-Suppes scott
  • Conjecture 1.10: Stanley-Stembridge stanleystembridge
  • ...and 33 more