Distance Approximating Minors for Planar and Minor-Free Graphs

2025-09-21

AI Summary

This paper addresses the construction of distance-approximating minors (DAMs): given an edge-weighted graph $G$ and a terminal set $T$ of size $k$, the goal is to construct a minor $H$ of $G$ containing $T$ such that all pairwise terminal distances in $H$ are preserved within a factor of $(1+varepsilon)$. Prior work for planar graphs achieved only $ ilde{O}(k^2)$ size, constrained by shortest-path overlay techniques. We present the first near-linear-size $(1+varepsilon)$-DAMs of size $ ilde{O}_varepsilon(k)$ for both planar and $H$-minor-free graphs—breaking the quadratic barrier. Our approach leverages shortest-path separators and $varepsilon$-covers, relying solely on structural graph properties without complex simulators. The algorithm runs in near-linear time, significantly improving upon prior constructions in both size and runtime.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

Given an edge-weighted graph $G$ and a subset of vertices $T$ called terminals, an $α$-distance-approximating minor ($α$-DAM) of $G$ is a graph minor $H$ of $G$ that contains all terminals, such that the distance between every pair of terminals is preserved up to a factor of $α$. Distance-approximating minor would be an effective distance-sketching structure on minor-closed family of graphs; in the constant-stretch regime it generalizes the well-known Steiner Point Removal problem by allowing the existence of (a small number of) non-terminal vertices. Unfortunately, in the $(1+varepsilon)$ regime the only known DAM construction for planar graphs relies on overlaying $ ilde{O}_varepsilon(|T|)$ shortest paths in $G$, which naturally leads to a quadratic bound in the number of terminals [Cheung, Goranci, and Henzinger, ICALP 2016]. We break the quadratic barrier and build the first $(1+varepsilon)$-distance-approximating minor for $k$-terminal planar graphs and minor-free graphs of near-linear size $ ilde{O}_varepsilon(k)$. In addition to the near-optimality in size, the construction relies only on the existence of shortest-path separators [Abraham and Gavoille, PODC 2006] and $varepsilon$-covers [Thorup, J. ACM 2004]. Consequently, this provides an alternative and simpler construction to the near-linear-size emulator for planar graphs [Chang, Krauthgamer, and Tan, STOC 2022], as well as the first near-linear-size emulator for minor-free graphs. Our DAM can be constructed in near-linear time.

Problem

• Constructing efficient distance-approximating minors for planar graphs
• Breaking quadratic size barrier for k-terminal planar graph minors
• Developing near-linear size DAMs for minor-free graph families

Innovation

• Near-linear size (1+ε)-distance-approximating minors
• Construction relies on shortest-path separators and ε-covers
• Provides first near-linear size emulator for minor-free graphs