[论文解读] Bi-Arc Digraphs and Conservative Polymorphisms

Pavol Hell, Arash Rafiey+1 more|arXiv (Cornell University)|Aug 11, 2016

Advanced Graph Theory Research参考文献 46被引用 7

一句话总结

本文引入了双弧有向图作为区间图的广泛推广，并为它们提供了多项式时间的识别算法。它证明了双弧有向图恰好是那些允许保守半格多对映射的有向图，提供了禁止障碍物的刻画，从而解决了该类别的长期未解的识别问题。主要贡献在于对具有保守半格多对映射的关系结构的识别复杂性的完整二分法分类，表明只有当所有关系为一元关系，最多一个二元关系时，该问题才是多项式时间可解的。

ABSTRACT

In this paper we study the class of bi-arc digraphs, important from two seemingly unrelated perspectives. On the one hand, they are precisely the digraphs that admit certain polymorphisms of interest in the study of constraint satisfaction problems; on the other hand, they are a very broad generalization of interval graphs. Bi-arc digraphs is the class of digraphs that admit conservative semilattice polymorphisms. There is much interest in understanding structures that admit particular types of polymorphisms, and especially in their recognition algorithms. (Such problems are referred to as metaproblems.) Surprisingly, the class of bi-arc digraphs also describes the class of digraphs that admit certain other kinds of conservative polymorphisms. Thus solving the recognition problem for bi-arc digraphs solves the metaproblem for digraphs for several types of conservative polymorphisms. The complexity of the recognition problem for digraphs with conservative semilattice polymorphisms was an open problem, while it was known to be NP-complete for certain more complex relational structures. We complement our result by providing a complete dichotomy classification of which general relational structures have polynomial or NP-complete recognition problems for the existence of conservative semilattice polymorphisms. Bi-arc digraphs also generalizes the class of interval graphs; in fact it reduces to the class of interval graphs for symmetric and reflexive digraphs. It is much broader than interval graphs and includes other generalizations of interval graphs such as co-threshold tolerance graphs and adjusted interval digraphs. Yet, it is still a reasonable extension of interval graphs, in the sense that it keeps much of the appeal of interval graphs. Our main result is a forbidden obstruction characterization of, and a polynomial recognition for, the class of bi-arc digraphs.

研究动机与目标

• 刻画允许保守半格多对映射的有向图类。
• 为双弧有向图提供禁止障碍物刻画和多项式时间识别算法。
• 解决关系结构中保守半格多对映射的识别问题的复杂性。
• 将已知类如区间图、共阈值容忍图和调整后的区间有向图统一并推广到一个单一框架下。
• 建立具有保守半格多对映射的结构的识别复杂性的二分法分类。

提出的方法

• 使用一对有向图构造来分析顶点对及其结构约束。
• 引入 H+ 概念，并研究该辅助结构中的路径和环路以检测障碍物。
• 采用两阶段算法：首先通过弧约束识别潜在的最小排序，然后通过一致性检查进行细化。
• 对多对映射应用变换以强制实现最小排序行为，确保结合律和保守性。
• 通过将三元关系约化到更高元关系来证明当元数大于3时问题为 NP-完全。
• 基于四同余和四受限路径的禁止结构刻画来定义障碍物。

实验结果

研究问题

• RQ1 阻止有向图允许保守半格多对映射的完整障碍物集合是什么？
• RQ2 允许保守半格多对映射的有向图的识别问题是否可在多项式时间内求解？
• RQ3 双弧有向图与已知图类（如区间图和共阈值容忍图）有何关系？
• RQ4 对于哪些关系结构，保守半格多对映射的存在性可在多项式时间内判定？
• RQ5 允许保守半格多对映射的高元关系结构的识别复杂性是什么？

主要发现

• 双弧有向图类恰好是那些允许保守半格多对映射的有向图集合。
• 通过基于 H+ 中路径和环路分析的两阶段方法，为双弧有向图提供了多项式时间识别算法。
• 当且仅当所有关系为一元关系，最多一个二元关系时，保守半格多对映射的识别问题是多项式时间可解的。
• 对于元数大于三的关系结构，识别问题是 NP-完全的。
• 允许保守循环或多对称多对映射（所有元数）的有向图类与双弧有向图类完全一致。
• 本文提供了完整的二分法分类：当所有关系为一元关系，最多一个二元关系时，保守半格多对映射的识别问题是多项式时间可解的；否则为 NP-完全。