1658. Highway Dimension: a Metric View
Invited abstract in session MB-20: Integer Programming and Combinatorial Optimization: Complexity Questions and Algorithms, stream Combinatorial Optimization.
Monday, 10:30-12:00Room: Esther Simpson 2.11
Authors (first author is the speaker)
| 1. | Andreas Feldmann
|
| School of Computer Science, University of Sheffield |
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 ≈ 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.
Keywords
- Transportation
- Combinatorial Optimization
- Complexity and Approximation
Status: accepted
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