线性邻域复杂度类中的时间最优APSP与矩阵乘法
Time-Optimal APSP and Matrix Multiplication in Classes of Linear Neighborhood Complexity
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中文总结 AI 辅助
该研究针对线性邻域复杂度类图，提出了时间最优的APSP与矩阵乘法算法，解决了相关领域的多个问题并扩展了结果，还给出了三角形、K₄、K₅检测的高效算法。
中文摘要 AI 辅助
线性邻域复杂度是图类上一种非常通用的结构假设，覆盖了绝大多数稀疏图类（如平面图、排除固定（拓扑）子式的图、有界扩张图），以及许多结构化稠密图类（如有界团宽图、孪生宽图、合并宽图、翻转宽图）。本研究针对来自线性邻域复杂度类的n顶点图，为以下问题提出了时间复杂度为O(n²)的最优算法：1. 所有点对最短路径（APSP）；2. 输入图的邻接矩阵M与任意n×n矩阵的乘法。更具体地说，经过二次预处理后，我们可以在O(n)时间内将M与任意n向量相乘。这一成果解决了[Bonnet、Kim、Geniet、Moon；ICALP '26]中提出的若干问题，改进并推广了其他多篇近期论文的结果，包括[Bonnet、Giocanti、Ossona de Mendez、Thomassé；STACS '23]、[Bannach、Marwitz、Tantau；STACS '24]、[Anand、van den Brand、McCarty；NeurIPS '26]、[Kozma、Opler '26]以及[Cardinal、McCarty、Yuditsky '26]的成果。我们还将结果扩展到有界VC密度类。在线性邻域复杂度类中，我们还提出了三角形检测算法，其在n顶点m边图上的随机时间复杂度为线性时间O(n+m)；K₄检测算法的随机时间复杂度为O(n log⁵n + m log n)或确定性时间复杂度为O(n²)；K₅检测算法的随机时间复杂度为O(n log⁹n + m log⁵n)。
英文摘要
The notion of linear neighborhood complexity is a very general structural assumption on a graph class, covering most classes of sparse graphs such as planar graphs, graphs excluding a fixed (topological) minor, or bounded expansion graphs, as well as many structured classes of dense graphs, such as graphs of bounded clique-width, twin-width, merge-width, or flip-width. In this work, we present $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: $\bullet$ All-Pairs Shortest Paths, $\bullet$ the multiplication of the adjacency matrix $M$ of the input graph with any $n \times n$ matrix. More specifically, after a quadratic preprocessing, we can multiply $M$ with any $n$-vector in $O(n)$ time. This solves several questions raised in [Bonnet, Kim, Geniet, Moon; ICALP '26], and improves and generalizes results in several other recent papers [Bonnet, Giocanti, Ossona de Mendez, Thomassé; STACS '23], [Bannach, Marwitz, Tantau; STACS '24], [Anand, van den Brand, McCarty; NeurIPS '26], [Kozma, Opler '26], and [Cardinal, McCarty, Yuditsky '26]. We also extend our results to classes of bounded VC density. In classes of linear neighborhood complexity, we also give a triangle-detection algorithm in randomized linear time $O(n+m)$ in $n$-vertex $m$-edge graphs, a $K_4$-detection algorithm in randomized $O(n \log^5 n + m \log n)$ or deterministic $O(n^2)$ time, and a $K_5$-detection algorithm in randomized $O(n \log^9 n + m \log^5 n)$ time.
