[论文解读] Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings

Marek Cygan, Harold N. Gabow+1 more|arXiv (Cornell University)|Apr 7, 2012

Advanced Graph Theory Research参考文献 29被引用 11

一句话总结

本论文提出了一种通用的算法框架，利用Baur-Strassen定理和Strojohann的行列式算法，解决具有整数边权且无负权圈的图中的最短圈、直径、半径以及最小权完美匹配问题。该框架在所有这些问题上实现了$\tilde{O}(Wn^{\tau})$的时间复杂度，为具有负权边的无向图提供了首个此类高效解法，并解决了Sankowski（2006）提出的关于一般图中最小权完美匹配的开放问题。

ABSTRACT

Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen's theorem and of Strojohann's determinant algorithm. It allows us to give new and simple solutions to the following problems: * Finding Shortest Cycles -- We give a simple ilde{O}(Wn^ω) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with non-negative weights) this matches the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the other hand, no algorithm working in ilde{O}(Wn^ω) time was previously known for undirected graphs with negative weights. Furthermore our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. * Computing Diameter and Radius -- We give a simple ilde{O}(Wn^ω) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge no algorithm with this running time was known for undirected graphs with negative weights. * Finding Minimum Weight Perfect Matchings -- We present an ilde{O}(Wn^ω) time algorithm for finding minimum weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski in 2006, who presented such an algorithm but only in the case of bipartite graphs. In order to solve minimum weight perfect matching problem we develop a novel combinatorial interpretation of the dual solution which sheds new light on this problem. Such a combinatorial interpretation was not know previously, and is of independent interest.

研究动机与目标

• 开发一种统一的代数框架，利用矩阵乘法求解加权图中的基本最短路径与匹配问题。
• 解决无向图中负权边下最短圈与完美匹配问题缺乏高效算法的问题。
• 解决Sankowski（2006）提出的关于在一般无向图中计算最小权完美匹配的开放问题。
• 将现有技术扩展至计算具有负权边图的直径与半径，此前方法仅限于非负权边。

提出的方法

• 利用Baur-Strassen定理计算多项式矩阵行列式的偏导数，从而高效计算最短圈权重。
• 使用Strojohann算法在多项式环上快速计算行列式。
• 在有限域上应用随机代入以评估符号导数，并检测非零项以判断圈或路径的成员关系。
• 通过顶点拆分将无向图中寻找最短圈的问题转化为$n$个最小权完美匹配计算。
• 结合基于行列式的代数计算与二分查找，以$\tilde{O}(Wn^{\tau})$时间计算直径与半径。
• 提出最小权完美匹配中对偶解的新组合解释，提供了新的结构洞见。

实验结果

研究问题

• RQ1 能否基于矩阵乘法构建统一的代数框架，用于求解具有负权边的图中的多个基本图问题？
• RQ2 能否在$\tilde{O}(Wn^{\tau})$时间内计算无向图中具有负权边的最短圈？
• RQ3 能否在$\tilde{O}(Wn^{\tau})$时间内求解一般（非二分）无向图中的最小权完美匹配问题，从而解决2006年提出的开放问题？
• RQ4 能否使用代数方法在$\tilde{O}(Wn^{\tau})$时间内计算具有负权边图的直径与半径？
• RQ5 能否利用符号行列式计算高效地计算位于长度至多为$t$的圈上的顶点集合？

主要发现

• 本论文首次提出$\tilde{O}(Wn^{\tau})$时间复杂度的算法，用于求解具有负权边的无向图中的最短圈问题。
• 首次提出$\tilde{O}(Wn^{\tau})$时间复杂度的算法，用于计算具有负权边的无向图的直径与半径。
• 通过给出$\tilde{O}(Wn^{\tau})$时间复杂度的算法，解决了Sankowski于2006年提出的开放问题，即一般无向图中最小权完美匹配的计算问题。
• 该框架在$\tilde{O}(Wn^{\tau})$时间内检测负权圈，达到与有向图中最佳已知界限相同的性能。
• 该算法在$\tilde{O}(Wn^{\tau})$时间内计算位于长度至多为$t$的圈上的顶点集合，优于先前的$\tilde{O}(Wn^{\tau}t)$时间复杂度。
• 提出了最小权完美匹配中对偶解的新组合解释，具有独立的理论价值。