Separator Theorems for Minor-Free and Shallow Minor-Free Graphs with Applications

Alon, Seymour, and Thomas generalized Lipton and Tarjan's planar separator theorem and showed that a $K_h$-minor free graph with $n$ vertices has a separator of size at most $h^{3/2}\sqrt n$. They gave an algorithm that, given a graph $G$ with $m$ edges and $n$ vertices and given an integer $h\geq 1$, outputs in $O(\sqrt{hn}m)$ time such a separator or a $K_h$-minor of $G$. Plotkin, Rao, and Smith gave an $O(hm\sqrt{n\log n})$ time algorithm to find a separator of size $O(h\sqrt{n\log n})$. Kawarabayashi and Reed improved the bound on the size of the separator to $h\sqrt n$ and gave an algorithm that finds such a separator in $O(n^{1 + \epsilon})$ time for any constant $\epsilon > 0$, assuming $h$ is constant. This algorithm has an extremely large dependency on $h$ in the running time (some power tower of $h$ whose height is itself a function of $h$), making it impractical even for small $h$. We are interested in a small polynomial time dependency on $h$ and we show how to find an $O(h\sqrt{n\log n})$-size separator or report that $G$ has a $K_h$-minor in O(\poly(h)n^{5/4 + \epsilon}) time for any constant $\epsilon > 0$. We also present the first O(\poly(h)n) time algorithm to find a separator of size $O(n^c)$ for a constant $c < 1$. As corollaries of our results, we get improved algorithms for shortest paths and maximum matching. Furthermore, for integers $\ell$ and $h$, we give an $O(m + n^{2 + \epsilon}/\ell)$ time algorithm that either produces a $K_h$-minor of depth $O(\ell\log n)$ or a separator of size at most $O(n/\ell + \ell h^2\log n)$. This improves the shallow minor algorithm of Plotkin, Rao, and Smith when $m = \Omega(n^{1 + \epsilon})$. We get a similar running time improvement for an approximation algorithm for the problem of finding a largest $K_h$-minor in a given graph.Comment: To appear at FOCS 201

Single source shortest paths in H-minor free graphs

AbstractWe present an algorithm for the Single Source Shortest Paths (SSSP) problem in directed H-minor free graphs. For every fixed H, if G is a graph with n vertices having integer edge lengths and s is a designated source vertex of G, the algorithm runs in Õ(n11.5−2logL)≤O(n1.392logL) time, where L is the absolute value of the smallest edge length. The algorithm computes the shortest paths and the distances from s to all vertices of the graph, or else provides a certificate that G is not H-minor free. Our result improves an earlier O(n1.5logL) time algorithm for this problem, which follows from a general SSSP algorithm of Goldberg