Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles

TL;DR

This paper introduces three new algorithms for finding graph separators in shallow minor-free graphs, improving efficiency and applicability through novel use of decremental approximate distance oracles.

Contribution

The paper presents three algorithms with improved running times and separator sizes for shallow minor-free graphs, utilizing a novel application of decremental approximate distance oracles.

Findings

First algorithm achieves $O( ext{poly}(h)\, ext{ell}\, m^{1+ ext{epsilon}})$ time.

Second algorithm attains $O( ext{poly}(h)(\, ext{sqrt} ext{ell}\, n^{1+ ext{epsilon}} + n^{2+ ext{epsilon}}/ ext{ell}^{3/2}))$ time.

Third algorithm runs in $O( ext{poly}(h) ext{sqrt} ext{ell}\, n^{1+ ext{epsilon}})$ time with a specific separator size.

Abstract

Plotkin, Rao, and Smith (SODA'97) showed that any graph with $m$ edges and $n$ vertices that excludes $K_h$ as a depth $O(\ell\log n)$-minor has a separator of size $O(n/\ell + \ell h^2\log n)$ and that such a separator can be found in $O(mn/\ell)$ time. A time bound of $O(m + n^{2+\epsilon}/\ell)$ for any constant $\epsilon > 0$ was later given (W., FOCS'11) which is an improvement for non-sparse graphs. We give three new algorithms. The first has the same separator size and running time $O(\mbox{poly}(h)\ell m^{1+\epsilon})$. This is a significant improvement for small $h$ and $\ell$. If $\ell = \Omega(n^{\epsilon'})$ for an arbitrarily small chosen constant $\epsilon' > 0$, we get a time bound of $O(\mbox{poly}(h)\ell n^{1+\epsilon})$. The second algorithm achieves the same separator size (with a slightly larger polynomial dependency on $h$) and running time $O(\mbox{poly}(h)(\sqrt\ell n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2}))$ when $\ell = \Omega(n^{\epsilon'})$. Our third algorithm has running time $O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon})$ when $\ell = \Omega(n^{\epsilon'})$. It finds a separator of size $O(n/\ell) + \tilde O(\mbox{poly}(h)\ell\sqrt n)$ which is no worse than previous bounds when $h$ is fixed and $\ell = \tilde O(n^{1/4})$. A main tool in obtaining our results is a novel application of a decremental approximate distance oracle of Roditty and Zwick.