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Nested and outlier embeddings into trees

Shuchi Chawla, Kristin Sheridan

TL;DR

This work investigates instance-optimal probabilistic embeddings of finite metrics into hierarchically separated trees (HSTs) while allowing outliers. By developing probabilistic nested compositions and a robust merge framework for HSTs, the authors show how to sample embeddings that keep distortion near a target $c$ on most points while discarding at most $O( rac{k}{ ext{ε}} ext{log}^2 k)$ outliers, achieving $(32+ ext{ε})c$ distortion. The core technical advances include non-contractive, probabilistic nested embeddings into HSTs, a perfect merge operation for HSTs, and a modified LP rounding approach to identify and handle outliers, plus extensions to weighted outliers. The framework yields improved instance-specific approximations for Buy-at-Bulk, Dial-A-Ride, and related network-design problems, highlighting practical gains in metrics where removing a small subset of points dramatically improves embedability into trees or ultrametrics.

Abstract

In this paper, we consider outlier embeddings into HSTs and ultrametrics. In particular, for $(X,d)$, let $k$ be the size of the smallest subset of $X$ such that all but that subset (i.e. the ``outlier set'') can be probabilistically embedded into the space of HSTs with expected distortion at most $c$. Our primary result is showing that there exists an efficient algorithm that takes in $(X,d)$ and a target distortion $c$ and samples from a probabilistic embedding with at most $O(\frac k ε\log^2k)$ outliers and distortion at most $(32+ε)c$, for any $ε>0$. This leads to better instance-specific approximations for certain instances of the buy-at-bulk and dial-a-ride problems, whose current best approximation algorithms go through HST embeddings. In order to facilitate our results, we largely focus on the concept of compositions of nested embeddings introduced by [Chawla and Sheridan 2024]. A nested embedding is a composition of two embeddings of a metric space $(X,d)$ -- a low distortion embedding of a subset $S$ of nodes, and a higher distortion embedding of the entire metric. The composition is a single embedding that preserves the low distortion over $S$ and does not increase distortion over the remaining points by much. In this paper, we expand this concept from the setting of deterministic embeddings to the setting of probabilistic embeddings. We show how to find good nested compositions of embeddings into HSTs, and combine this with an approximation algorithm of [Munagala et al. 2023] to obtain our results.

Nested and outlier embeddings into trees

TL;DR

This work investigates instance-optimal probabilistic embeddings of finite metrics into hierarchically separated trees (HSTs) while allowing outliers. By developing probabilistic nested compositions and a robust merge framework for HSTs, the authors show how to sample embeddings that keep distortion near a target on most points while discarding at most outliers, achieving distortion. The core technical advances include non-contractive, probabilistic nested embeddings into HSTs, a perfect merge operation for HSTs, and a modified LP rounding approach to identify and handle outliers, plus extensions to weighted outliers. The framework yields improved instance-specific approximations for Buy-at-Bulk, Dial-A-Ride, and related network-design problems, highlighting practical gains in metrics where removing a small subset of points dramatically improves embedability into trees or ultrametrics.

Abstract

In this paper, we consider outlier embeddings into HSTs and ultrametrics. In particular, for , let be the size of the smallest subset of such that all but that subset (i.e. the ``outlier set'') can be probabilistically embedded into the space of HSTs with expected distortion at most . Our primary result is showing that there exists an efficient algorithm that takes in and a target distortion and samples from a probabilistic embedding with at most outliers and distortion at most , for any . This leads to better instance-specific approximations for certain instances of the buy-at-bulk and dial-a-ride problems, whose current best approximation algorithms go through HST embeddings. In order to facilitate our results, we largely focus on the concept of compositions of nested embeddings introduced by [Chawla and Sheridan 2024]. A nested embedding is a composition of two embeddings of a metric space -- a low distortion embedding of a subset of nodes, and a higher distortion embedding of the entire metric. The composition is a single embedding that preserves the low distortion over and does not increase distortion over the remaining points by much. In this paper, we expand this concept from the setting of deterministic embeddings to the setting of probabilistic embeddings. We show how to find good nested compositions of embeddings into HSTs, and combine this with an approximation algorithm of [Munagala et al. 2023] to obtain our results.
Paper Structure (26 sections, 18 theorems, 20 equations, 3 figures, 7 algorithms)

This paper contains 26 sections, 18 theorems, 20 equations, 3 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Nested compositions into HSTs with bounded contraction.
  3. Probabilistic nested embedding.
  4. Definitions and results
  5. Applications
  6. Bad metrics
  7. Buy-at-Bulk
  8. Dial-A-Ride
  9. Minimum cost communication tree with outliers
  10. Probabilistic nested compositions of embeddings
  11. Nested and outlier embeddings for HSTs
  12. Compositions of nested embeddings for HSTs
  13. Outlier embeddings for HSTs
  14. Bounding the optimal LP solution
  15. Rounding the LP
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(X,d)$ be any metric that admits a $(k,c)$ probabilistic outlier embedding into HSTs. Then for any constant $\epsilon>0$, there exists an algorithm that efficiently samples from a probabilistic embedding into HSTs of some $S\subseteq X$ with $|X\setminus S|\leq O(\frac{1}{\epsilon}k\log^2k)$ an

Figures (3)

  • Figure 1: We can consider an unweighted graph on $n$ nodes such that one part of the graph is $K_{n-\log n}$ (the complete graph on $n-\log n$ nodes), and the remaining $\log n$ nodes form some constant-degree expander graph $G_{\log n}$. The two parts of the graph can then be connected by a single edge. Note that the subgraph induced by each of these two sub-parts of this graph is an isometric subgraph of the original graph.
  • Figure 2: A visual example of metric composition where $M$ is some metric on the set $\set{x,y,z}$.
  • Figure 3: This figure depicts what happens in the algorithm $\texttt{MergeHST}$. In particular, $u$ is the node common to both HSTs on the left side of the figure. The triangles represent subtrees, and the letters below the triangles indicate the presence of the corresponding node as a leaf of that subtree. Note that after merging trees, the least common ancestor of $x$ and $y$ is that of $x$ and $u$ from the first tree, as this was higher than that of $y$ and $u$. Further, the least common ancestor of $x'$ and $y'$ is at the height of the least common ancestor of $u$ and $y'$ in the second tree, as that is higher than the least common ancestor of $u$ and $x'$ in the first tree. This is the foundational idea behind why $d_{\alpha}(x,y)\geq \max\set{d_{\alpha_1}(u,x),d_{\alpha_2}(u,y)}$ holds when $\alpha$ is the output embedding for $\texttt{MergeHST}$ on inputs $\alpha_1$ and $\alpha_2$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: Probabilistic composition of nested embeddings, modified from chawla2024
  • Definition 2.3
  • Theorem 2.4: HSTs have strong nested embeddings
  • Theorem 2.5
  • Corollary 2.6
  • Definition 3.1: Metric composition bartal2003
  • Theorem 3.2
  • Theorem 3.3
  • ...and 35 more