A $2^{O(k)}n$ algorithm for $k$-cycle in minor-closed graph families

TL;DR

This paper introduces a randomized algorithm for finding cycles of length k in minor-closed graph families with a runtime of 2^{O(k)}n, improving upon previous super-exponential dependence on k, and also provides a derandomized version.

Contribution

It presents the first fixed-parameter algorithm with single-exponential dependence on k for cycle detection in minor-closed graph families, applicable to both directed and undirected graphs.

Findings

Runs in 2^{O(k)}n time for randomized version.

Derandomized version runs in 2^{O(k)}n log n time.

Applicable to both directed and undirected graphs.

Abstract

Let ${\mathcal C}$ be a proper minor-closed family of graphs. We present a randomized algorithm that given a graph $G \in {\mathcal C}$ with $n$ vertices, finds a simple cycle of size $k$ in $G$ (if exists) in $2^{O(k)}n$ time. The algorithm applies to both directed and undirected graphs. In previous linear time algorithms for this problem, the runtime dependence on $k$ is super-exponential. The algorithm can be derandomized yielding a $2^{O(k)}n\log n$ time algorithm.