A graph is chordal if every cycle of length greater than three contains an edge between non-adjacent vertices. Chordal graphs are of interest both theoretically, since they admit polynomial time solutions to a range of NP-hard graph problems, and practically, since they arise in many applications including sparse linear algebra, computer vision, and computational biology. A maximal chordal subgraph is a chordal subgraph that is not a proper subgraph of any other chordal subgraph. Existing algorithms for computing maximal chordal subgraphs depend on dynamically ordering the vertices, which is an inherently sequential process and therefore limits the algorithms' parallelizability. In this paper we explore techniques to develop a scalable parallel algorithm for extracting a maximal chordal subgraph. We demonstrate that an earlier attempt at developing a parallel algorithm may induce a non-optimal vertex ordering and is therefore not guaranteed to terminate with a maximal chordal subgraph. We then give a new algorithm that first computes and then repeatedly augments a spanning chordal subgraph. After proving that the algorithm terminates with a maximal chordal subgraph, we then demonstrate that this algorithm is more amenable to parallelization and that the parallel version also terminates with a maximal chordal subgraph. That said, the complexity of the new algorithm is higher than that of the previous parallel algorithm, although the earlier algorithm computes a chordal subgraph which is not guaranteed to be maximal. We experimented with our augmentation-based algorithm on both synthetic and real-world graphs. We provide scalability results and also explore the effect of different choices for the initial spanning chordal subgraph on both the running time and on the number of edges in the maximal chordal subgraph.

中文翻译：

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一种新的基于扩充的最大和弦子图提取算法。

如果每个长度大于3的循环在非相邻顶点之间包含边，则该图为弦。弦图在理论上很有意义，因为它们允许多项式时间解解决一系列NP硬图问题，而在实践上，因为它们出现在许多应用中，包括稀疏线性代数，计算机视觉和计算生物学。最大和弦子图是一个和弦子图，不是任何其他和弦子图的适当子图。用于计算最大弦子图的现有算法取决于顶点的动态排序，这是固有的顺序过程，因此限制了算法的可并行性。在本文中，我们探索了开发可伸缩并行算法以提取最大弦子图的技术。我们证明了开发并行算法的较早尝试可能会导致非最佳顶点排序，因此不能保证以最大弦子图终止。然后，我们提供了一种新算法，该算法首先计算然后重复扩展跨度弦子图。在证明算法以最大弦子图终止后，我们将证明该算法更适合并行化，并且并行版本也以最大弦子图终止。也就是说，新算法的复杂度高于先前的并行算法，尽管较早的算法会计算不保证最大的弦子图。我们在合成图和现实图上都试验了基于增广的算法。