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arXiv 2605.14902math.COcs.DMcs.DS
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k-不相交路径问题的最优界

Optimal Bounds for the k-Disjoint Paths Problem

Technische Universität Berlin（柏林工业大学）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究所离散数学组）；LIRMM, Univ Montpellier, CNRS（蒙彼利埃大学CNRS LIRMM实验室）；School of Computing, KAIST（韩国科学技术院计算机学院）

Dario Cavallaro, Maximilian Gorsky, Stephan Kreutzer, Dimitrios M. Thilikos, Sebastian Wiederrecht

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AI总结：

本文提出了一种通用的无关顶点定理，用于k-不相交路径问题和根小世界检查的共同推广，即(k,d)-folio问题。通过引入bidimensionality的概念，改进了算法参数依赖性，并优化了图小世界算法框架的量化表现。

摘要：

图 minors 系列为算法结构图论奠定了基础，为如不相交路径、根小世界检查和 folio 等问题提供了固定参数算法。k-不相交路径问题的固定参数可解性背后的关键成分是无关顶点技术。该工具由 Vital Linkage 定理和所谓的 Linkage 函数 ℓ 所支配。然而，尽管其基础作用，已知的 Linkage 函数的界限仍然庞大且仅隐式理解。这些结果背后的量化界限传统上如此之大，以至于所得到的算法被视为“银河级”的。我们的主要结果是一个通用的无关顶点定理，用于 k-不相交路径和根小世界检查的共同推广，即 (k,d)-folio 问题。具体来说，我们证明了对于任何图 G，其中 k 个终端选择自某个集合 R，如果 G 的树宽超过 β(k,b,d)∈2^{poly(b + d)}⋅poly(k)，则可以定位 (k,d)-folio 问题的无关顶点。此处，量 b 是 R 的 bidimensionality，即 G 中能根在 R 上的 (b×b) 网格 minor 的最大 b。因此，无关顶点阈值的指数部分由 bidimensionality 的界限驱动，而非终端数量。我们论证这种依赖性本质上在多项式因子内是最佳的。作为结果，Linkage 函数满足 ℓ(k)∈2^{poly(k)}。除了其结构意义外，我们的结果还为不相交路径和根小世界检查的算法提供了改进的参数依赖性，并为广泛的图小世界算法框架提供了量化改进。

英文摘要：

The Graph Minors Series of Robertson and Seymour forms the foundation of algorithmic structural graph theory, yielding fixed-parameter algorithms for problems such as Disjoint Paths, Rooted Minor Checking, and Folio. A key ingredient behind the fixed-parameter tractability of the $k$-Disjoint Paths problem is the irrelevant-vertex technique. This machinery is governed by the Vital Linkage Theorem and the so-called Linkage Function $\ell$. However, despite its foundational role, the best known bounds on the Linkage Function are enormous and are only implicitly understood. The quantitative bounds behind these results have traditionally been so large that the resulting algorithms are regarded as "galactic". Our main result is a general irrelevant-vertex theorem for a common generalisation of $k$-Disjoint Paths and Rooted Minor Checking for graphs of size at most $d,$ commonly called the $(k,d)$-Folio problem. Specifically, we show that for any graph $G$ in which the $k$ terminals are chosen from some set $R,$ if the treewidth of $G$ exceeds $β(k,b,d)\in$ $2^{{\bf poly}(b + d)}$ $\cdot {\bf poly}(k)$ then we can locate an irrelevant vertex for the $(k,d)$-Folio problem. Here, the quantity $b$ is the bidimensionality of $R,$ that is, the largest $b$ for which a $(b\times b)$-grid minor in $G$ can be rooted on $R$. Thus, the exponential component of the irrelevant-vertex threshold is driven by the bound on the bidimensionality, rather than by the number of terminals, and we argue that this dependence is essentially optimal up to polynomial factors. As a consequence, the Linkage Function satisfies $\ell(k) \in 2^{{\bf poly}(k)}$. Beyond its structural significance, our result yields improved parameter dependencies for algorithms for Disjoint Paths and Rooted Minor Checking}, and provides a quantitative improvement for a broad range of graph-minor-based algorithmic frameworks.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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