We consider the parameterised $k,e$-Long Cycle problem, in which you are given an $n$-vertex undirected graph $G$, a specified edge $e$ in $G$, and a positive integer $k$, and are asked to decide if the graph $G$ has a simple cycle through $e$ of length at least $k$. We show how to solve the problem in $1.731^k\operatorname{poly}(n)$ time, improving over the $2^k\operatorname{poly}(n)$ time algorithm by [Fomin et al., TALG 2024], but not the more recent $1.657^k\operatorname{poly}(n)$ time algorithm by [Eiben, Koana, and Wahlstr\"om, SODA 2024]. When the graph is bipartite, we can solve the problem in $2^{k/2}\operatorname{poly}(n)$ time, matching the fastest known algorithm for finding a cycle of length exactly $k$ in an undirected bipartite graph [Bj\"orklund et al., JCSS 2017]. Our results follow the approach taken by [Fomin et al., TALG 2024], which describes an efficient algorithm for finding cycles using many colours in a vertex-coloured undirected graph. Our contribution is twofold. First, we describe a new algorithm and analysis for the central colourful cycle problem, with the aim of providing a comparatively short and self-contained proof of correctness. Second, we give tighter reductions from $k,e$-Long Cycle to the colourful cycle problem, which lead to our improved running times.

翻译：我们研究参数化的 $k,e$-长环问题：给定一个 $n$ 顶点无向图 $G$、图中一条指定边 $e$ 以及一个正整数 $k$，需要判断图 $G$ 是否存在一条经过边 $e$ 且长度至少为 $k$ 的简单环。我们展示了如何在 $1.731^k\operatorname{poly}(n)$ 时间内解决该问题，改进了 [Fomin 等人, TALG 2024] 提出的 $2^k\operatorname{poly}(n)$ 时间算法，但尚未超越 [Eiben, Koana 和 Wahlström, SODA 2024] 近期提出的 $1.657^k\operatorname{poly}(n)$ 时间算法。当图为二分图时，我们可以在 $2^{k/2}\operatorname{poly}(n)$ 时间内解决问题，这与在无向二分图中寻找长度恰好为 $k$ 的环的最快已知算法 [Björklund 等人, JCSS 2017] 性能相当。我们的结果遵循了 [Fomin 等人, TALG 2024] 所采用的方法，该方法描述了一种在顶点着色的无向图中利用多色寻找环的高效算法。我们的贡献有两个方面。首先，我们为核心的多彩环问题提出了一种新算法与分析，旨在提供一个相对简洁且自包含的正确性证明。其次，我们给出了从 $k,e$-长环问题到多彩环问题的更紧规约，从而实现了我们改进的运行时间。