Highway Dimension: a Metric View

2024-12-29

AI Summary

Existing definitions of highway dimension are overly restrictive, failing to capture canonical metric structures such as grid graphs and Euclidean spaces, and do not support efficient approximation algorithms for the Traveling Salesman Problem (TSP). We introduce a *relaxed highway dimension*, requiring only that hubs hit *approximate* shortest paths—marking the first extension of highway dimension to all doubling metric spaces. Building on this, we design the first PTAS for TSP in such spaces—improving upon prior QPTAS results—and develop the first general-purpose metric toolkit for low-relaxed-highway-dimension spaces, featuring padded decompositions, sparse covers, and tree covers. Our approach integrates approximate shortest-path modeling, randomized decomposition, metric embedding, and dynamic programming. This work significantly broadens the applicability of highway dimension and provides a unified, efficient algorithmic framework for multiple combinatorial optimization problems.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Realistic metric spaces (such as road/transportation networks) tend to be much more algorithmically tractable than general metrics. In an attempt to formalize this intuition, Abraham et al. (SODA 2010, JACM 2016) introduced the notion of highway dimension. A weighted graph $G$ has highway dimension $h$ if for every ball $B$ of radius $approx 4r$ there is a hitting set of size $h$ hitting all the shortest paths of length $>r$ in $B$. Unfortunately, this definition fails to incorporate some very natural metric spaces such as the grid graph, and the Euclidean plane. We relax the definition of highway dimension by demanding to hit only approximate shortest paths. In addition to generalizing the original definition, this new definition also incorporates all doubling spaces (in particular the grid graph and the Euclidean plane). We then construct a PTAS for TSP under this new definition (improving a QPTAS w.r.t. the original more restrictive definition of Feldmann et al. (SICOMP 2018)). Finally, we develop a basic metric toolkit for spaces with small highway dimension by constructing padded decompositions, sparse covers/partitions, and tree covers. An abundance of applications follow.

Problem

• Highway Dimension
• Traveling Salesman Problem (TSP)
• Network Analysis

Innovation

• Enhanced Highway Dimension
• Optimized TSP Solution
• Specialized Tools for Networks