Near-Linear $\varepsilon$-Emulators for Planar Graphs

TL;DR

This paper introduces near-linear size $ ext{(1+} ext{ extvarepsilon)}$-emulators for planar graphs that efficiently approximate distances between terminal vertices, significantly improving size bounds and enabling faster algorithms for key graph problems.

Contribution

The authors construct the first near-linear size $ ext{ extvarepsilon}$-emulators for planar graphs, breaking previous quadratic bounds and enabling efficient approximation algorithms.

Findings

Emulators of size $ ilde O(k/ ext{ extvarepsilon}^{O(1)})$ for planar graphs.

Breakthrough in size bounds compared to previous quadratic bounds.

Facilitate fast approximation algorithms for shortest paths, cuts, diameter, and dynamic distances.

Abstract

We study vertex sparsification for distances, in the setting of planar graphs with distortion: Given a planar graph $G$ (with edge weights) and a subset of $k$ terminal vertices, the goal is to construct an $\varepsilon$-emulator, which is a small planar graph $G'$ that contains the terminals and preserves the distances between the terminals up to factor $1+\varepsilon$. We construct the first $\varepsilon$-emulators for planar graphs of near-linear size $\tilde O(k/\varepsilon^{O(1)})$. In terms of $k$, this is a dramatic improvement over the previous quadratic upper bound of Cheung, Goranci and Henzinger, and breaks below known quadratic lower bounds for exact emulators (the case when $\varepsilon=0$). Moreover, our emulators can be computed in (near-)linear time, which lead to fast $(1+\varepsilon)$-approximation algorithms for basic optimization problems on planar graphs, including multiple-source shortest paths, minimum $(s,t)$-cut, graph diameter, and dynamic distace oracle.