[论文解读] Contributions to the Problems of Recognizing and Coloring Gammoids

Immanuel Albrecht|arXiv (Cornell University)|Jan 1, 2018

Graph Labeling and Dimension Problems参考文献 41被引用 3

一句话总结

本文通过引入一种新颖的复杂度度量，定义了在余因子和对偶性下封闭的子类，提出了一种纯粹组合的方法，无需使用幂级数即可从有向图表示计算R-矩阵表示，并证明了所有格路径拟阵都是3-可着色的。此外，本文还提供了一个全面的算法，通过α-违反准则识别拟阵，并提出了一种新的拟阵定向方法。

ABSTRACT

This work provides a thorough introduction to the field of gammoids and presents new results that are considered helpful for solving the problems of recognizing and coloring gammoids. Matroids are set systems that generalize the concept of linear independence between sets of rows of a matrix over a field. Gammoids are those matroids that may be represented by directed graphs where the corresponding independence is modeled as the existence of certain families of pair-wise vertex disjoint paths. The seminal papers in gammoid theory have been written by J.H. Mason [2], A.W. Ingleton and M.J. Piff [1]. Natural applications of gammoids can be found within the realms of connectivity of both directed and undirected graphs. In this work, we introduce our concept of the complexity of a gammoid, which may be used to define subclasses of the class of gammoids that inherit the most notable properties of the class of gammoids: being closed under minors, duality, and direct sums. Furthermore, we provide a comprehensive method for deciding whether a given matroid is a gammoid. We give a new procedure for obtaining an R-matrix, that represents a gammoid given by the means of a directed graph, which avoids using power series. We present the first purely combinatorial way of obtaining orientations of gammoids. We prove that every lattice path matroid is 3-colorable. In Chapter 1 we give a brief introduction to matroid theory: we present axiomatizations of matroids most relevant to this work, the concepts of minors and duality as well as representability over fields and properties of extensions. The same chapter also contains a brief introduction to the theory of transversals, including the Theorems of Hall, Rado, Ore, and Perfect, and an introduction to transversal matroids. Also, we provide a short introduction to directed graphs, we introduce the concept of a routing in a directed graph and we close the chapter with Menger’s Theorem and its consequences. In Chapter 2 we define gammoids as matroids that may be obtained from routings in directed graphs. We explore the properties of their directed graph representations and along that we define our notion of a duality respecting representation which correlates the duality-like notion of opposite directed graphs with the notion of duality with respect to gammoids. Furthermore, we introduce our three complexity measures for gammoids that yield subclasses of gammoids which are closed under minors and duality. We present Mason’s α-criterion for strict gammoids, and we examine the properties of strict gammoids and transversal matroids. We analyze the problem of recognizing gammoids, we develop the notion of an α-violation, and we present our best approach for deciding instances of the recognition problem. At the end of Chapter 2, we present our method for determining an R-matrix representing a gammoid from a given representation in terms of a directed graph. In Chapter 3 we shortly introduce oriented matroids and their associated concept of colorings. We show that all orientations of lattice path matroids have 3-colorings. Then we introduce our concept of a heavy arc orientation of a gammoid that yields a purely combinatorial way to obtain representable orientations of gammoids. In Chapter 4 we summarize our new results and give an overview of new and old open problems. [1] A.W. Ingleton and M.J. Piff. Gammoids and transversal matroids. Journal of Combinatorial Theory, Series B, 15(1):51–68, 1973. [2] J.H. Mason. On a class of matroids arising from paths in graphs. Proceedings of the London Mathematical Society, 3(1):55–74, 1972.

研究动机与目标

• 开发一种新的拟阵复杂度度量，以定义在余因子和对偶性下封闭的子类。
• 提供一种纯粹组合的方法，从有向图表示计算拟阵的R-矩阵表示，避免使用幂级数。
• 建立一个全面的决策程序，用于判断给定的拟阵是否为拟阵。
• 提出一种新的组合方法，用于定向拟阵。
• 证明所有格路径拟阵都是3-可着色的。

提出的方法

• 引入三种拟阵复杂度度量，可生成在余因子和对偶性下封闭的子类。
• 开发一种新方法，从有向图表示获得拟阵的R-矩阵表示，无需使用幂级数。
• 提出一种纯粹组合的方法用于定向拟阵，避免依赖代数或分析技术。
• 应用Mason的α-准则刻画严格拟阵，并分析其性质。
• 利用α-违反的概念指导拟阵识别算法。
• 运用Menger定理和有向图中的路由概念，形式化拟阵表示。

实验结果

研究问题

• RQ1 如何正式定义拟阵的复杂度，以生成在余因子和对偶性下封闭的子类？
• RQ2 能否从有向图表示组合计算拟阵的R-矩阵表示，而无需使用幂级数？
• RQ3 保证一个拟阵是拟阵的最小条件集是什么？如何实现算法化检测？
• RQ4 是否存在一种纯粹组合的方法用于定向拟阵，独立于代数或分析构造？
• RQ5 所有格路径拟阵是否都是3-可着色的？其何种结构特性支持这一性质？

主要发现

• 证明了所有格路径拟阵都是3-可着色的，确立了这一重要拟阵子类的强结构性质。
• 开发了一种新的组合方法，从有向图表示计算拟阵的R-矩阵表示，避免使用幂级数。
• 本文引入了一种拟阵复杂度度量，可生成在余因子和对偶性下封闭的子类，从而实现对拟阵家族的更精细分类。
• 提出了一套全面的算法，用于判断给定拟阵是否为拟阵，其基础是检测α-违反。
• 引入了一种新颖的、纯粹组合的拟阵定向程序，提供了一种独立于代数或分析工具的构造性方法。
• 形式化了对偶性尊重表示的概念，将拟阵中的对偶性与相反有向图联系起来。