An AI enhanced approach to the tree unimodality conjecture

TL;DR

The paper investigates the tree unimodality conjecture through the lens of log-concavity of the independence sequence, using PatternBoost to actively search for counterexamples. It encodes trees via Prüfer codes and trains a transformer-guided local-global search to maximize a score indicating log-concavity failure at a chosen index, primarily around $N/2$. The study reports tens of thousands of new counterexamples for trees with $N$ ranging from $27$ to $101$, revealing diverse structural forms and highlighting the influence of the independence number on search dynamics. It also analyzes the limitations of the approach, including a tendency toward path-like trees and the critical role of path-punishment techniques to uncover counterexamples at $N/2-1$, while offering insight into future directions for broader index exploration and global sequence shaping. The findings demonstrate both the power and boundaries of AI-assisted discovery in extremal combinatorics and raise questions about the prevalence and distribution of non-log-concave independence sequences in large trees.

Abstract

Given a graph $G$, its independence sequence is the integral sequence $a_1,a_2,...,a_n$, where $a_i$ is the number of independent sets of vertices of size i. In the late 80's Alavi, Erdos, Malde, Schwenk showed that this sequence need not be unimodal for general graphs, but conjectured that it is always unimodal whenever $G$ is a tree. This conjecture was then naturally generalized to claim that the independence sequence of trees should be log concave, in the sense that $a_i^2$ is always above $a_{i-1}a_{i+1}$. This conjecture stood for many years, until in 2023, Kadrawi, Levit, Yosef, and Mizrachi proved that there were exactly two trees on 26 vertices whose independence sequence was not log concave. In this paper, we use the AI architecture PatternBoost, developed by Charton, Ellenberg, Wagner, and Williamson to train a machine to find counter-examples to the log-concavity conjecture. We will discuss the successes of this approach - finding tens of thousands of new counter-examples to log-concavity with vertex set sizes varying from 27 to 101 - and some of its fascinating failures.

Figures (17)

• Figure 1: The two counter-examples, $T_1 = \mathbf{3_{44}}$ and $T_2 =\mathbf{3^{*}_{34}}$ to log-concavity with 26 vertices. These images have been taken from their source Kadrawi2023OnComputing
• Figure 2: An example of a single edge swap
• Figure 3: The top 50,000 scores in each epoch during our first attempts to find breakage at $N/2-1$. Here, $N = 56$
• Figure 4: The histogram of top scores in generation 6, and generation 7, respectively, when looking for log-concavity breakage at index 27 for trees of order 56. The histogram for the 6th generation is quite similar looking to those for the 2nd through 5th generations as well.
• Figure :

Theorems & Definitions (24)

• Definition 2.1
• Conjecture 2.2: The Tree Unimodality Conjecture, Alavi1987VertexIndependence
• Definition 2.3
• Remark 2.4
• Definition 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 4.1
• Conjecture 4.2