[论文解读] Computational Complexity of Covering Multigraphs with Semi-Edges: Small Cases

Jan Bok, Jiřı́ Fiala+3 more|arXiv (Cornell University)|Jan 1, 2021

Advanced Graph Theory Research参考文献 33被引用 1

一句话总结

本文首次对包含半边（仅与一个顶点相连的边）的多重图中的图覆盖问题展开计算复杂性分析，表明半边通过要求显式的边映射（而不仅顶点映射）显著增加了问题的难度。本文全面分类了含半边的一至两个顶点的多重图的覆盖问题复杂性，证明即使在二分图输入下问题仍为NP难，仅在特定条件下（如目标图为单顶点且输入为二分图）才存在多项式时间可解性。

ABSTRACT

We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for graphs with semi-edges. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello et al. asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show that the presence of semi-edges makes the covering problem considerably harder; e.g., it is no longer sufficient to specify the vertex mapping induced by the covering, but one necessarily has to deal with the edge mapping as well. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding k-tuples of pairwise disjoint perfect matchings in regular graphs, or finding equitable partitions of regular bipartite graphs).

研究动机与目标

• 启动对包含半边（在拓扑图论和数学物理中日益重要的边，仅与一个顶点相连）的多重图中图覆盖问题计算复杂性的研究。
• 研究半边如何改变图覆盖问题的复杂性，相较于仅含边的经典图，特别在于其要求除顶点映射外还需进行边映射。
• 对含半边的一至两个顶点的多重图覆盖问题提供完整的计算复杂性分类。
• 通过证明新特征化适用于最多两个度数大于2的顶点的简单目标图的无限类，强化现有无半边覆盖问题的结果。
• 探索图覆盖与边着色问题之间的联系，特别通过在归约中使用(b,b)-着色。

提出的方法

• 从3-SAT问题归约，构造图G，使得当且仅当原始3-SAT公式可满足时，G存在(b,b)-着色，使用特定顶点与边结构的变量与子句构件。
• 设计每个变量对应2b+2个顶点的变量构件，通过K1与K2子图连接，利用结构约束强制所有变量出现位置的颜色一致。
• 构造子句构件，使得较小部分中每个顶点的度数为2b，当且仅当子句中恰好b个变量被设为真时，其恰好有b个红色与b个蓝色邻居。
• 将(b,b)-着色用作有效覆盖投射的代理，其中颜色分配对应覆盖图中的边与顶点映射。
• 通过反证法证明，在任意有效(b,b)-着色中，所有变量的出现必须具有相同颜色，利用邻域约束与度数平衡。
• 将结果推广至更广泛的情形，包括正则图中存在k元两两不相交完美匹配对的情况，支持主要复杂性结论。

实验结果

研究问题

• RQ1 与仅含边的经典图相比，半边的引入如何影响图覆盖问题的计算复杂性？
• RQ2 在允许半边的情况下，一顶点与两顶点多重图的覆盖问题的计算复杂性是什么？
• RQ3 即使输入图是二分图，半边的存在是否仍能使覆盖问题变为NP难？
• RQ4 是否存在关于含半边覆盖问题何时为多项式时间可解的结构或组合特征化？
• RQ5 无半边覆盖问题的结果在多大程度上可推广至含半边的情形，特别是在复杂性分类方面？

主要发现

• 含半边的一顶点多重图的覆盖问题，若输入图为二分图，则为多项式时间可解，但一般情况下仍为NP难。
• 对于含半边的两顶点目标多重图，即使输入图是二分图，也无法保证可解性，且对于正则二分图输入，问题仍为NP难。
• 本文建立了含半边的一至两个顶点多重图覆盖问题的完整复杂性分类，解决了覆盖问题层级中一个非平凡的情形。
• 即使对于简单的输入图，覆盖问题的NP难性也已得到证明；在正则两顶点目标情况下，该难性在二分图输入下依然成立。
• 在构造图G中(b,b)-着色的特征化与有效覆盖投射之间建立了双射关系，从而实现从3-SAT到覆盖问题的归约。
• 结果通过将特征化应用于最多两个度数大于2的顶点的简单目标图的无限类，强化了无半边覆盖问题的既有结论。