[论文解读] Computing directed path-width and directed tree-width of recursively defined digraphs

Frank Gurski, Carolin Rehs|arXiv (Cornell University)|Jun 12, 2018

Advanced Graph Theory Research参考文献 29被引用 4

一句话总结

本文通过为不相交并集、顺序复合、有向并集和系列复合建立构造性分解规则，证明了在递归定义的有向图（包括有向补图）中，有向路径宽与有向树宽相等，从而提出了一种线性时间算法来计算这两类参数。主要贡献在于将无向补图中已知的路径宽与树宽相等的结果推广到有向情形，使得能够高效计算一大类有向图的宽度参数。

ABSTRACT

In this paper we consider the directed path-width and directed tree-width of recursively defined digraphs. As an important combinatorial tool, we show how the directed path-width and the directed tree-width can be computed for the disjoint union, order composition, directed union, and series composition of two directed graphs. These results imply the equality of directed path-width and directed tree-width for all digraphs which can be defined by these four operations. This allows us to show a linear-time solution for computing the directed path-width and directed tree-width of all these digraphs. Since directed co-graphs are precisely those digraphs which can be defined by the disjoint union, order composition, and series composition our results imply the equality of directed path-width and directed tree-width for directed co-graphs and also a linear-time solution for computing the directed path-width and directed tree-width of directed co-graphs, which generalizes the known results for undirected co-graphs of Bodlaender and Moehring.

研究动机与目标

• 将无向补图中路径宽与树宽相等的已知结果推广到有向情形。
• 为递归定义的有向图开发一种构造性方法，以计算有向路径宽与有向树宽。
• 将无向补图的结果推广到有向补图及扩展的有向补图。
• 为有向补图中的这两类宽度参数提供一种线性时间算法。
• 通过分解操作，为有向图中的宽度参数建立理论基础。

提出的方法

• 本文在有向图上定义了四种操作——不相交并集、顺序复合、有向并集和系列复合，以递归构建复杂的有向图。
• 证明了对于每种操作，结果有向图的有向路径宽与有向树宽均可由其组成部分的参数计算得出。
• 提供了构造性证明，以从各组成部分的分解构建有向路径分解与有向树分解。
• 引入了双向完全子有向图引理与二分图子有向图引理，以支持分解规则。
• 证明了所有由这四种操作构造的有向图，其有向路径宽与有向树宽均相等。
• 通过证明有向补图与扩展有向补图在四种操作下封闭，将该方法扩展至有向补图与扩展有向补图。

实验结果

研究问题

• RQ1 能否通过分解规则高效计算递归定义有向图的有向路径宽与有向树宽？
• RQ2 对于通过不相交并集、顺序复合、有向并集和系列复合构造的有向图，有向路径宽与有向树宽是否仍保持相等？
• RQ3 有向补图中，有向路径宽与有向树宽的相等性是否保持成立，从而推广无向补图的结果？
• RQ4 能否为有向补图中的两类宽度参数构造一种线性时间算法？
• RQ5 有向图的宽度参数与其强连通分量及无环缩聚图的参数之间有何关系？

主要发现

• 所有通过不相交并集、顺序复合、有向并集和系列复合构造的有向图，其有向路径宽与有向树宽均相等。
• 对于此类有向图（包括有向补图），存在一种线性时间算法，可同时计算有向路径宽与有向树宽。
• 对于有向补图，有向路径宽与有向树宽等于最大双定向团大小减一。
• 一般有向图的有向路径宽与有向树宽受其强连通分量中这些参数最大值的限制。
• 本研究推广了Bodlaender与Möhri针对无向补图的先前工作，将路径宽与树宽的相等性推广至有向情形。
• 该方法对有向树宽的不同定义具有鲁棒性，包括允许分解中出现空集的定义，但Reed的定义可能产生更高的值。