Finding longer cycles via shortest colourful cycle

TL;DR

The paper addresses the parameterised $k,e$-Long Cycle problem by reducing it to a colourful-cycle problem and develops a new, self-contained algorithm for Shortest $k,t,e$-Colourful Cycle that runs in $O(2^k\operatorname{poly}(n))$ time using a determinant-based algebraic framework in a field of characteristic $2$. This method yields an overall $1.731^k\operatorname{poly}(n)$ time algorithm for $k,e$-Long Cycle in general undirected graphs and a matching $2^{k/2}\operatorname{poly}(n)$ time algorithm for bipartite graphs, via tight, structure-aware reductions. The approach centers on encoding many-colour cycles through labelled cycle covers and evaluating a carefully constructed polynomial $g$ as a sum of permanents that, under random sieving, isolates the desired cycle length. The key contributions are a simpler, self-contained algorithmic proof of correctness for the colourful-cycle approach, and tighter reductions from $k,e$-Long Cycle to this colourful-cycle problem, enabling the improved running times with both general and bipartite graph considerations. This advances fixed-parameter tractable techniques for long cycles and clarifies the role of colour-based algebraic methods in cycle detection, potentially impacting nearby problems like $k$-Long paths and Steiner-like cycle problems.

Abstract

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 Wahlström, 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 [Björklund 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.

Theorems & Definitions (28)

• Theorem 1: Long cycle in bipartite undirected graphs
• Theorem 2: Main; Long cycle in undirected graphs
• Theorem 3: Shortest colourful cycle
• Lemma 1
• Lemma 2
• proof
• Lemma 3: Running time
• proof
• Lemma 4: Label sum for subgraphs
• proof