[论文解读] Directed Width Measures and Monotonicity of Directed Graph Searching

Łukasz Kaiser, Stephan Kreutzer+2 more|arXiv (Cornell University)|Aug 20, 2014

Advanced Graph Theory Research参考文献 14被引用 4

一句话总结

该论文通过分析警察与强盗游戏，建立了一个有向图宽度度量的全面层次结构，证明了有界凯利宽度蕴含有界DAG宽度，并表明在有向树宽度游戏中，警察单调性代价是无界的——解决了有向图结构理论中长期存在的开放问题。

ABSTRACT

We consider generalisations of tree width to directed graphs, that attracted much attention in the last fifteen years. About their relative strength with respect to "bounded width in one measure implies bounded width in the other" many problems remain unsolved. Only some results separating directed width measures are known. We give an almost complete picture of this relation. For this, we consider the cops and robber games characterising DAG-width and directed tree width (up to a constant factor). For DAG-width games, it is an open question whether the robber-monotonicity cost (the difference between the minimal numbers of cops capturing the robber in the general and in the monotone case) can be bounded by any function. Examples show that this function (if it exists) is at least $f(k) > 4k/3$ (Kreutzer, Ordyniak 2008). We approach a solution by defining weak monotonicity and showing that if $k$ cops win weakly monotonically, then $O(k^2)$ cops win monotonically. It follows that bounded Kelly-width implies bounded DAG-width, which has been open since the definition of Kelly-width by Hunter and Kreutzer in 2008. For directed tree width games we show that, unexpectedly, the cop-monotonicity cost (no cop revisits any vertex) is not bounded by any function. This separates directed tree width from D-width defined by Safari in 2005, refuting his conjecture.

研究动机与目标

• 澄清DAG宽度、凯利宽度、D-宽度和有向树宽度等有向宽度度量之间的相对强度。
• 解决有界凯利宽度是否蕴含有界DAG宽度这一开放问题。
• 研究有向图搜索游戏中单调性的代价，特别是通用策略与单调策略之间的差距。
• 确定有向树宽度是否严格弱于D-宽度，从而反驳文献中的一项猜想。
• 提供各种有向宽度参数之间关系的近乎完整图景。

提出的方法

• 分析DAG宽度和有向树宽度的警察与强盗游戏，重点关注单调性性质。
• 引入弱单调性作为连接弱单调策略与强单调策略的工具。
• 证明：若k名警察能弱单调获胜，则O(k²)名警察能单调获胜，从而建立二次上界。
• 使用树分解和有向树分解技术，关联宽度度量并证明不等式。
• 通过边定向和树分解中的路径覆盖，运用反证法证明不可能性结果。
• 利用已知的有向树宽度与D-宽度结果，比较其相对强度与分离性。

实验结果

研究问题

• RQ1 有界凯利宽度是否蕴含有界DAG宽度？
• RQ2 在有向树宽度游戏中，警察单调性代价（单调策略与非单调策略所需警察数量之差）是否无界？
• RQ3 D-宽度是否严格强于有向树宽度，还是二者等价？
• RQ4 DAG宽度游戏中的单调性代价是否可被任何函数有界？
• RQ5 凯利宽度与DAG宽度之间的精确关系是什么？

主要发现

• 有界凯利宽度蕴含有界DAG宽度，解决了自凯利宽度定义以来长期存在的开放问题。
• 有向树宽度游戏中警察单调性代价无界，证明有向树宽度严格弱于D-宽度。
• 对于DAG宽度游戏，若k名警察能弱单调获胜，则O(k²)名警察能单调获胜，建立了二次上界。
• 本文反驳了[Saf05]中的一个猜想，即D-宽度与有向树宽度等价，表明二者可分离。
• 建立了有向宽度度量的近乎完整层次结构，大多数情况下存在严格不等式。
• 结果表明，有向树宽度是所研究参数中最一般的宽度参数，DAG宽度和凯利宽度均位于其之下。