[论文解读] Acyclic coloring of special digraphs

Frank Gurski, Dominique Komander+1 more|arXiv (Cornell University)|Jun 24, 2020

Advanced Graph Theory Research参考文献 52被引用 4

一句话总结

本文提出了一种针对有向图的有向 clique-width 为常数时计算色数的首个多项式时间算法，基于有向 clique-width 表达式树的自底向上动态规划。关键贡献在于：针对有向 clique-width 参数化时的 XP 算法，以及在同时以有向 clique-width 和颜色数为参数时的 FPT 算法；此外，对有向 cograph 及其推广类，也给出了线性时间解法。

ABSTRACT

An acyclic r-coloring of a directed graph G=(V,E) is a partition of the vertex set V into r acyclic sets. The dichromatic number of a directed graph G is the smallest r such that G allows an acyclic r-coloring. For symmetric digraphs the dichromatic number equals the well-known chromatic number of the underlying undirected graph. This allows us to carry over the W[1]-hardness and lower bounds for running times of the chromatic number problem parameterized by clique-width to the dichromatic number problem parameterized by directed clique-width. We introduce the first polynomial-time algorithm for the acyclic coloring problem on digraphs of constant directed clique-width. From a parameterized point of view our algorithm shows that the Dichromatic Number problem is in XP when parameterized by directed clique-width and extends the only known structural parameterization by directed modular width for this problem. Furthermore, we apply defineability within monadic second order logic in order to show that Dichromatic Number problem is in FPT when parameterized by the directed clique-width and r. For directed co-graphs, which is a class of digraphs of directed clique-width 2, and several generalizations we even show linear time solutions for computing the dichromatic number. Furthermore, we conclude that directed co-graphs and the considered generalizations lead to subclasses of perfect digraphs. For directed cactus forests, which is a set of digraphs of directed tree-width 1, we conclude an upper bound of 2 for the dichromatic number and we show that an optimal acyclic coloring can be computed in linear time.

研究动机与目标

• 为特殊类有向图的色数问题开发高效算法。
• 解决有向图中无环着色的计算复杂性问题，特别是针对具有有界结构参数的图类。
• 通过有向 clique-width 将无向图着色的参数化复杂性结果推广至有向图。
• 证明当以有向 clique-width 和颜色数为参数时，色数问题属于 FPT。
• 识别出完美有向图的子类（如有向 cograph 和有向 cactus 森林），其色数可被高效计算。

提出的方法

• 使用输入有向图的有向 clique-width 表达式所导出的有根表达式树进行自底向上的动态规划。
• 对每个子树，通过每种无环颜色类的可达性对多重集维护基于标签的可达性信息。
• 计算使每个颜色类诱导出无环子有向图所需的最少颜色数。
• 应用一阶逻辑（MSO）可定义性，证明当以有向 clique-width 和颜色数为参数时，问题属于 FPT。
• 利用有向 cograph（clique-width 为 2）和有向 cactus 森林（tree-width 为 1）的已知结构性质，推导出线性时间算法。
• 将无环着色问题转化为具有 O(n²) 个变量的整数规划，以证明当以顶点数为参数时，问题属于 FPT。

实验结果

研究问题

• RQ1 对于有向 clique-width 为常数的有向图，色数问题是否可在多项式时间内求解？
• RQ2 当以有向 clique-width 和颜色数为参数时，色数问题是否为固定参数可追踪（FPT）？
• RQ3 有向 cograph 及其推广类是否具有计算色数的线性时间算法？
• RQ4 当以有向 clique-width 为参数时，色数问题是否可在 XP 时间内求解？
• RQ5 当以底层无向图的 clique-width 为参数时，色数问题是否属于 FPT？

主要发现

• 本文首次提出针对有向 clique-width 为常数的有向图的色数问题的多项式时间算法，时间复杂度为 n^{2^{O(k²)}}，其中 k 为有向 clique-width。
• 通过 MSO 可定义性，证明当以有向 clique-width 和颜色数为参数时，色数问题属于 FPT。
• 对于有向 cograph 及其推广类（有向 clique-width 为 2），色数可在 O(n) 时间内计算。
• 有向 cactus 森林（有向树宽为 1）的色数至多为 2，且最优无环着色可在 O(n) 时间内计算。
• 当以底层无向图的 clique-width 为参数时，该问题不属于 FPT，除非 P = NP。
• 当以颜色数 r 为参数时，该问题不属于 XP，因为 DCN₂ 是 NP-完全的。