[论文解读] Digraph Decompositions and Monotonicity in Digraph Searching

Stephan Kreutzer, Sebastian Ordyniak|ArXiv.org|Feb 15, 2008

Advanced Graph Theory Research参考文献 17被引用 22

一句话总结

本文通过证明在有向图上，针对隐形且迟钝的贼或可见且快速的贼的两种版本的警察与强盗游戏，其搜索策略均非单调，从而解决了有向图搜索领域长期存在的开放性问题。作者构造了明确的反例，表明单调策略所需警察数量远超非单调策略；进一步表明，即使在有向图宽度假设极低（如DAG-width极小）的情况下，许多自然问题仍保持NP完全，从而限制了有向图分解在算法上的应用范围。

ABSTRACT

We consider monotonicity problems for graph searching games. Variants of these games - defined by the type of moves allowed for the players - have been found to be closely connected to graph decompositions and associated width measures such as path- or tree-width. Of particular interest is the question whether these games are monotone, i.e. whether the cops can catch a robber without ever allowing the robber to reach positions that have been cleared before. The monotonicity problem for graph searching games has intensely been studied in the literature, but for two types of games the problem was left unresolved. These are the games on digraphs where the robber is invisible and lazy or visible and fast. In this paper, we solve the problems by giving examples showing that both types of games are non-monotone. Graph searching games on digraphs are closely related to recent proposals for digraph decompositions generalising tree-width to directed graphs. These proposals have partly been motivated by attempts to develop a structure theory for digraphs similar to the graph minor theory developed by Robertson and Seymour for undirected graphs, and partly by the immense number of algorithmic results using tree-width of undirected graphs and the hope that part of this success might be reproducible on digraphs using a directed tree-width. Unfortunately the number of applications for the digraphs measures introduced so far is still small. We therefore explore the limits of the algorithmic applicability of digraph decompositions. In particular, we show that various natural candidates for problems that might benefit from digraphs having small directed tree-width remain NP-complete even on almost acyclic graphs.

研究动机与目标

• 解决有向图上警察与强盗游戏在存在隐形且迟钝的贼时是否具有单调性这一长期开放问题。
• 探究有向图分解在算法可解性方面的极限，特别是与DAG-width等宽度度量的关系。
• 确定诸如图Grundy编号和核问题等自然问题在有向图有小有向宽度时是否仍具有可解性。
• 确立这些游戏中单调性代价的无界性，从而挑战单调策略中常数因子近似解的可行性。

提出的方法

• 通过设计特定的有向图构造明确的反例，以证明两种游戏变体的非单调性。
• 将已知的NP完全问题（如3-SAT、FAS、图Grundy编号、核问题）归约至有界DAG-width的有向图，以证明其不可解性。
• 为特定子图（如变量部件和子句环）设计单调获胜策略，以控制DAG-width。
• 通过分析图的结构成分（如环、无环子图）证明所构造图的DAG-width至多为2。
• 运用博弈论推理表明，非单调策略在所需警察数量方面可优于单调策略。
• 利用现有通过警察与强盗游戏对宽度度量的刻画，将游戏的单调性与分解理论相联系。

实验结果

研究问题

• RQ1 在有向图上，针对隐形且迟钝的贼的警察与强盗游戏是否具有单调性？
• RQ2 在有向图上，针对可见且快速的贼的警察与强盗游戏是否具有单调性？
• RQ3 在有界DAG-width的有向图上，图Grundy编号和核问题能否在多项式时间内求解？
• RQ4 是否存在这些有向图游戏单调性代价（即所需额外警察数量）的常数上界？
• RQ5 有向图分解在多大程度上能为除连通性和哈密顿性之外的自然问题提供可解算法？

主要发现

• 在有向图上，针对隐形且迟钝的贼的警察与强盗游戏是非单调的，已通过明确的反例证明。
• 在有向图上，针对可见且快速的贼的警察与强盗游戏同样是非单调的，类似反例表明其单调性代价无界。
• 反馈弧集（FAS）问题在DAG-width至多为四的有向图上仍是NP完全的。
• 图Grundy编号问题在DAG-width至多为二的有向图上仍是NP完全的。
• 核问题在DAG-width为二的有向图上仍是NP完全的。
• 即使在近乎无环的有向图（如DAG-width ≤ 2）上，许多自然问题仍保持NP完全，表明有向宽度度量在算法能力上存在根本性限制。