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 that the problem can be solved in $1.731^k\operatorname{poly}(n)$ time, improving over the previously best known $2^k\operatorname{poly}(n)$ time algorithm and solving an open problem [Fomin et al., TALG 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 $G$ 中的一条指定边 $e$ 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。我们证明该问题可在 $1.731^k\operatorname{poly}(n)$ 时间内求解，改进了先前已知的 $2^k\operatorname{poly}(n)$ 时间算法，并解决了[Fomin et al., TALG 2024]中提出的一个开放性问题。当图为二分图时，我们可在 $2^{k/2}\operatorname{poly}(n)$ 时间内求解该问题，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 et al., JCSS 2017]的时间复杂度相匹配。我们的结果遵循了[Fomin et al., 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种颜色高效寻找环的算法。我们的贡献主要体现在两个方面：首先，针对核心的多彩环问题，我们提出了一种新算法并进行了分析，旨在提供一个相对简洁且自洽的正确性证明；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。