Faster separators for shallow minor-free graphs via dynamic approximate distance oracles

Abstract

Plotkin, Rao, and Smith (SODA'97) showed that any graph with m edges and n vertices that excludes Kh as a depth O(ℓlogn)-minor has a separator of size O(n/ℓ+ℓh2logn) and that such a separator can be found in O(mn/ℓ) time. A time bound of O(m+n2+ε /ℓ) for any constant ε>0 was later given (W., FOCS'11) which is an improvement for non-sparse graphs. We give three new algorithms. The first two have the same separator size (the second having a slightly larger dependency on h) and running time O(poly(h)ℓn1+ε) and O(poly(h) (√ℓn1+ε + n2+ε/ℓ3/2)), respectively. The former is significantly faster than previous bounds for small h and ℓ. Our third algorithm has running time O(poly(h)(√ℓn 1+ε). It finds a separator of size O(n/ℓ) + Õ(poly(h)(ℓ√n) which is no worse than previous bounds when h is fixed and ℓ = Õ(n1/4. A main tool in obtaining our results is a decremental approximate distance oracle of Roditty and Zwick. © 2014 Springer-Verlag.