发表机构
The University of Tokyo; Institute for Chemical Reaction Design and Discovery, Hokkaido University（东京大学; 北海道大学化学反应设计与发现研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出首个线性拟阵奇偶性的最短增广路径算法，综合Gabow-Stallmann与Micali-Vazirani技术，将复杂度从O(nr^3)降至O(nr^2 log r)，并可进一步优化至O(nr^2)。
AI 中文摘要
拟阵奇偶性问题是一个基本框架，它同时推广了图匹配和拟阵交集问题。尽管一般版本是难解的，Lovász（1981）在线性拟阵奇偶性问题上开发了一个多项式时间算法，前提是假设矩阵表示可用。随后，Gabow和Stallmann（1986）提出了一种增广路径算法，该算法长期以来被认为是确定性算法中最快的之一。由于最短增广路径改进了图匹配（Micali & Vazirani, 1980）和线性拟阵交集（Cunningham, 1986）的算法，将这些技术扩展到线性拟阵奇偶性似乎是自然的进展。然而，这样的算法在四十年间一直难以实现。在本文中，我们提出了线性拟阵奇偶性的首个最短增广路径算法。我们的方法综合了Gabow和Stallmann的增广路径算法与Micali–Vazirani框架的同步花形成。我们的关键技术贡献有三方面：（i）一个线性代数论证，界定了线性拟阵奇偶性最短增广路径的长度，这推广了Cunningham对线性拟阵交集的界；（ii）通过长度下界对最短搜索路径的先验刻画；（iii）将Izumi、Kitamura和Yamaguchi（2025）建立的图匹配结构性质扩展到线性拟阵奇偶性设置。我们的算法在${\rm O}(nr^2\log r)$时间内确定性地解决线性拟阵奇偶性问题，其中$n$是基础集大小，$r$是拟阵秩。通过引入快速矩阵乘法，该复杂度可以进一步降低到${\rm O}(nr^2)$。这些结果改进了长期存在的确定性界${\rm O}(nr^3)$和${\rm O}(nr^\omega)$。
英文摘要
The matroid parity problem serves as a fundamental framework that generalizes both graph matching and matroid intersection. Although the general version is intractable, Lovász (1981) developed a polynomial-time algorithm for the linear matroid parity problem, assuming the availability of matrix representations. Subsequently, Gabow and Stallmann (1986) presented an augmenting path algorithm, which has long been recognized as one of the fastest deterministic algorithms. Since shortest augmenting paths improved algorithms for graph matching (Micali & Vazirani, 1980) and linear matroid intersection (Cunningham, 1986), extending these techniques to linear matroid parity appears to be a natural progression. However, such an algorithm has remained elusive for four decades. In this paper, we present the first shortest augmenting path algorithm for linear matroid parity. Our approach synthesizes the augmenting path algorithm of Gabow and Stallmann with the synchronized blossom formation of the Micali$\unicode{8211}$Vazirani framework. Our key technical contributions are threefold: (i) a linear-algebraic argument that bounds the lengths of shortest augmenting paths for linear matroid parity, which generalizes Cunningham's bound for linear matroid intersection; (ii) an a priori characterization of shortest search paths through lower bounds on their lengths; and (iii) an extension of the structural properties for graph matching established by Izumi, Kitamura, and Yamaguchi (2025) to the linear matroid parity setting. Our algorithm deterministically solves the linear matroid parity problem in ${\rm O}(nr^2\log r)$ time, where $n$ is the ground set size and $r$ is the matroid rank. By incorporating fast matrix multiplication, this complexity can be further reduced to ${\rm O}(nr^2)$. These results improve upon the long-standing deterministic bounds of ${\rm O}(nr^3)$ and ${\rm O}(nr^ω)$.
