Almost-linear ϵ-emulators for planar graphs

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 ϵ-emulator, which is a small planar graph G′ that contains the terminals and preserves the distances between the terminals up to factor 1+ϵ. We design the first ϵ-emulators for planar graphs of almost-linear size k1+o(1)/ϵ. In terms of k, this is a dramatic improvement over the previous quadratic upper bound of Cheung, Goranci and Henzinger [ICALP 2016], and breaks below known quadratic lower bounds for exact emulators (the case when ϵ=0). Moreover, our emulators can be computed in near-linear time, with applications to fast (1+ϵ)-approximation algorithms for basic optimization problems on planar graphs such as minimum (s,t)-cut and diameter.