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Getting to the Root of the Problem: Sums of Squares for Limits of Trees

Daniel Brosch, Diane Puges

TL;DR

This work extends inducibility from graphs to leaf-labeled rooted binary trees using Razborov's flag algebra framework combined with semidefinite programming. It develops a rooted-binary-tree flag algebra, formulates a SOS-based hierarchy, and computes hundreds of new inducibility bounds, including exact certificates for several small trees and the first outer approximations of tree-profiles. The authors demonstrate nonconvexity in some tree-profile regions and provide detailed examples such as I(E5)=2/3 with phantom-edge considerations. The approach yields a robust computational pipeline, delivering both numerical and rigorous results and enabling future extensions to other tree families and profile analyses.

Abstract

The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by Pippenger and Golumbic. In $2017$, Czabarka, Székely and Wagner extended this notion to leaf-labeled rooted binary trees, which are objects widely studied in the field of phylogenetics. They obtain the first results and bounds for the densities and inducibilities of such trees. Following up on their work, we apply Razborov's flag algebra theory to this setting, introducing the flag algebra of rooted leaf-labeled binary trees. This framework allows us to use polynomial optimization methods, based on semidefinite programming, to efficiently obtain new upper bounds for the inducibility of trees and to improve existing ones. Additionally, we obtain the first outer approximations of profiles of trees, which represent all possible simultaneous densities of a pair of trees in a sequence of trees of growing sizes. Finally, we are able to prove the non-convexity of some of these profiles.

Getting to the Root of the Problem: Sums of Squares for Limits of Trees

TL;DR

This work extends inducibility from graphs to leaf-labeled rooted binary trees using Razborov's flag algebra framework combined with semidefinite programming. It develops a rooted-binary-tree flag algebra, formulates a SOS-based hierarchy, and computes hundreds of new inducibility bounds, including exact certificates for several small trees and the first outer approximations of tree-profiles. The authors demonstrate nonconvexity in some tree-profile regions and provide detailed examples such as I(E5)=2/3 with phantom-edge considerations. The approach yields a robust computational pipeline, delivering both numerical and rigorous results and enabling future extensions to other tree families and profile analyses.

Abstract

The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in by Pippenger and Golumbic. In , Czabarka, Székely and Wagner extended this notion to leaf-labeled rooted binary trees, which are objects widely studied in the field of phylogenetics. They obtain the first results and bounds for the densities and inducibilities of such trees. Following up on their work, we apply Razborov's flag algebra theory to this setting, introducing the flag algebra of rooted leaf-labeled binary trees. This framework allows us to use polynomial optimization methods, based on semidefinite programming, to efficiently obtain new upper bounds for the inducibility of trees and to improve existing ones. Additionally, we obtain the first outer approximations of profiles of trees, which represent all possible simultaneous densities of a pair of trees in a sequence of trees of growing sizes. Finally, we are able to prove the non-convexity of some of these profiles.
Paper Structure (48 sections, 4 theorems, 95 equations, 23 figures, 3 tables)

This paper contains 48 sections, 4 theorems, 95 equations, 23 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Density and inducibility of rooted leaf-labeled binary trees
  3. Rooted leaf-labeled binary trees and subtrees
  4. Inducibility of trees
  5. Inducibility of graphs.
  6. Inducibility of leaf-labeled rooted binary trees.
  7. Previous results on the inducibility of trees.
  8. Tree profiles
  9. The flag algebra of rooted binary trees
  10. Model theoretic view
  11. Types and flags
  12. Densities of flags
  13. Products of flags
  14. Downward operator
  15. Flag sums of squares and semidefinite programming
  16. ...and 33 more sections

Key Result

Theorem 3.1

Let $n \geq 3$, and let $\mathcal{P}\subseteq [n]^3$ be a relation. We write $\mathcal{P}(i;j,k)$ for $(i,j,k)\in\mathcal{P}$. The two following statements are equivalent:

Figures (23)

  • Figure 1: Tree-profile of the caterpillar tree of size $4$ and the even tree of size $6$ with three known points and our conjecture for the upper boundary
  • Figure 2: All trees of sizes $3$, $4$ and $5$.
  • Figure 3: Subtree induced by the set of leaves $\{1,3,4\}$
  • Figure 4: Height of a leaf
  • Figure 5: $\mathrm{Cat}_{7}$
  • ...and 18 more figures

Theorems & Definitions (11)

  • Theorem 3.1
  • proof
  • Claim 3.1.1
  • proof
  • Claim 3.1.2
  • proof
  • Lemma 3.1.1: Chain rule for flags razborov2007
  • Theorem 3.2
  • Theorem 3.3: Theorem 3.1 in razborov2007
  • Remark 5.0.1
  • ...and 1 more