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.

💡 Research Summary

The paper revisits the classic separator theorems for graphs that exclude a complete graph $K_h$ as a minor and for graphs that exclude shallow (bounded‑depth) minors, and it dramatically improves both the size of the separators and the running time needed to compute them.
Alon, Seymour, and Thomas (1990) showed that any $K_h$‑minor‑free graph on $n$ vertices admits a separator of size $h^{3/2}\sqrt{n}$ and gave an $O(\sqrt{hn},m)$‑time algorithm to either find such a separator or produce a $K_h$‑minor. Later, Plotkin, Rao, and Smith (1997) obtained $O(h\sqrt{n\log n})$‑size separators in $O(hm\sqrt{n\log n})$ time, while Kawarabayashi and Reed (2010) reduced the size to $h\sqrt{n}$ but at the cost of an astronomically large dependence on $h$ (a power tower).
The present work eliminates the prohibitive $h$‑dependence and achieves polynomial‑in‑$h$ running times. Its main contributions are:

1. A unified algorithm that, for any constant $\varepsilon>0$, runs in $O(\mathrm{poly}(h),n^{5/4+\varepsilon})$ time and either returns a separator of size $O(h\sqrt{n\log n})$ or exhibits a $K_h$‑minor. This matches the best known separator size while keeping the runtime polynomial in $h$ and sub‑quadratic in $n$.

2. A linear‑time separator with sublinear size: the authors present the first $O(\mathrm{poly}(h),n)$‑time algorithm that produces a separator of size $O(n^{c})$ for any constant $c<1$ (specifically $c=4/5+\varepsilon$). Prior work could only achieve $c=2/3$ with linear time, and any $c<1$ required super‑linear time.

3. Improved shallow‑minor separators: for integers $\ell$ and $h$, they give an $O(m + n^{2+\varepsilon}/\ell)$‑time algorithm that either finds a $K_h$‑minor of depth $O(\ell\log n)$ or a separator of size $O(n/\ell + \ell h^{2}\log n)$. Choosing $\ell = \Theta(\sqrt{n}/(h\sqrt{\log n}))$ yields the $O(h\sqrt{n\log n})$ separator in $O(\mathrm{poly}(h),n^{5/4+\varepsilon})$ time, improving the Plotkin–Rao–Smith bound when $m = \Omega(n^{1+\varepsilon})$.

The technical core relies on two novel ideas. First, the algorithm maintains a dynamic $O(1)$‑stretch spanner of the yet‑unprocessed vertex set $V’$, which keeps the edge count low while preserving distances, allowing fast updates even when $m = O(n^{1+\varepsilon})$. Second, a “bootstrapping” technique builds a large separator quickly with a generous $\ell$, then uses it to construct an $r$‑clustering (clusters of size $O(r)$ with $O(h\sqrt{r})$ boundary vertices). By recursively clustering and applying the separator theorem inside each cluster, the overall runtime collapses to $O(\mathrm{poly}(h),n^{5/4+\varepsilon})$.

Beyond the separator results, the paper derives several algorithmic applications.

• Single‑source shortest paths (SSSP): For non‑negative edge weights, the new linear‑time separator yields an $O(\mathrm{poly}(h),n)$‑time SSSP algorithm in $K_h$‑minor‑free graphs, extending the planar‑graph result of Henzinger et al. (1997). For graphs with negative edge weights, the authors obtain an $\tilde O(\mathrm{poly}(h),n^{4/3}\log L)$ algorithm (where $L$ is the smallest absolute edge weight), improving on Yuster’s $O(\mathrm{poly}(h),n^{1.392}\log L)$ bound.
• Maximum matching: Using the new separator, a maximum matching can be computed in $O(\mathrm{poly}(h),n^{1.239})$ time, beating the previous $O(\mathrm{poly}(h),n^{1.326})$ result of Yuster and Zwick.

Overall, the paper establishes that for both minor‑free and shallow‑minor‑free graphs, one can simultaneously achieve near‑optimal separator size and truly sub‑quadratic (often near‑linear) construction time with only polynomial dependence on $h$. This bridges a long‑standing gap between theoretical separator bounds and practical algorithmic feasibility, and opens the door to faster algorithms for a wide range of graph problems in minor‑free families.