发表机构
Indian Institute of Technology Madras; University of Bergen（马德拉斯印度理工学院; 卑尔根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明边不相交最短环打包在参数k下是W[1]-难的，并引入更一般的多样最短环覆盖框架，在平面图上给出单指数算法，同时解决打包与多样性问题。
AI 中文摘要
Bentert、Fomin、Golovach、Korhonen、Lochet、Panolan、Ramanujan、Saurabh 和 Simonov（SODA 2025）开创了边不相交最短环打包的参数化研究：给定一个加权图 $G$ 和一个整数 $k$，判定 $G$ 是否包含 $k$ 个边不相交的最小权重环。他们证明了该问题存在一个运行时间为 $n^{O(k^6)}$ 的算法，并询问该问题在参数 $k$ 下是否为固定参数可解或 $W[1]$-难。我们通过证明边不相交最短环打包在参数 $k$ 下是 $W[1]$-难的来解决这个问题，即使在无权次立方图上也成立。同样的下界也适用于顶点不相交变体。对于平面图，他们提供了一个具有 $O(k^2)$ 个顶点的核的构造和一个运行时间为 $k^{O(k)} \cdot n^{O(1)}$ 的算法，并明确询问该问题是否承认一个运行时间为 $2^{O(k)} \cdot n^{O(1)}$ 的单指数算法。我们没有孤立地解决这个问题，而是引入了一个更一般的框架——多样最短环覆盖，它要求 $k$ 个最短环以受控方式重叠，同时最大化被覆盖边的总权重。该框架同时涵盖了边不相交最短环打包、寻找多样最短环的问题以及通过最短环最大化边覆盖的问题。我们的主要算法结果表明，多样最短环覆盖可以在平面图上以 $2^{O(k)} \cdot n^{O(1)}$ 的时间解决，从而作为特例给出了边不相交最短环打包的单指数算法。我们算法的关键思想是对层状最短环树进行结构分析，这是一种树状分解，揭示了平面图中最短环之间的层状交互模式，并实现了高效的动态规划算法。
英文摘要
Bentert, Fomin, Golovach, Korhonen, Lochet, Panolan, Ramanujan, Saurabh, and Simonov (SODA 2025) initiated the parameterized study of Edge-Disjoint Shortest Cycle Packing: given a weighted graph $G$ and an integer $k$, decide whether $G$ contains $k$ edge-disjoint cycles of minimum weight. They showed that the problem admits an algorithm running in time $n^{O(k^6)}$ and asked whether it is fixed-parameter tractable or $W[1]$-hard parameterized by $k$. We resolve this question by proving that Edge-Disjoint Shortest Cycle Packing is $W[1]$-hard parameterized by $k$, even on unweighted subcubic graphs. The same lower bound also applies to the vertex-disjoint variant. For planar graphs, they provides a construction of a kernel with $O(k^2)$ vertices and an algorithm running in time $k^{O(k)} \cdot n^{O(1)}$, and explicitly asked whether the problem admits a single-exponential algorithm of running time $2^{O(k)} \cdot n^{O(1)}$. Rather than addressing this question in isolation, we introduce a more general framework, Diverse Shortest Cycle Coverage, which asks for $k$ shortest cycles that may overlap in a controlled way while maximizing the total weight of covered edges. This framework simultaneously captures edge-disjoint shortest cycle packing, the problem of finding diverse shortest cycles, and the problem of maximizing edge coverage by shortest cycles. Our main algorithmic result shows that Diverse Shortest Cycle Coverage can be solved on planar graphs in time $2^{O(k)} \cdot n^{O(1)}$, thereby giving a single-exponential algorithm for Edge-Disjoint Shortest Cycle Packing as a special case. The key idea of our algorithm is a structural analysis of the Laminar Shortest Cycles Tree, a tree-like decomposition that reveals a laminar interaction pattern among shortest cycles in planar graphs and enables an efficient dynamic programming algorithm.
