We study a family of pattern-detection problems in vertex-colored temporal graphs. In particular, given a vertex-colored temporal graph and a multiset of colors as a query, we search for temporal paths in the graph that contain the colors specified in the query. These types of problems have several applications, for example in recommending tours for tourists or detecting abnormal behavior in a network of financial transactions. For the family of pattern-detection problems we consider, we establish complexity results and design an algebraic-algorithmic framework based on constrained multilinear sieving. We demonstrate that our solution scales to massive graphs with up to a billion edges for a multiset query with five colors and up to hundred million edges for a multiset query with ten colors, despite the problems being NP-hard. Our implementation, which is publicly available, exhibits practical edge-linear scalability and is highly optimized. For instance, in a real-world graph dataset with more than six million edges and a multiset query with ten colors, we can extract an optimum solution in less than eight minutes on a Haswell desktop with four cores.

中文翻译：

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使用代数指纹在大时间图中寻找路径图案

我们研究了顶点着色时间图中的一系列模式检测问题。特别是，给定一个顶点颜色的时间图和一组颜色作为查询，我们在图中搜索包含查询中指定颜色的时间路径。这些类型的问题有多种应用，例如为游客推荐旅游或检测金融交易网络中的异常行为。对于我们考虑的一系列模式检测问题，我们建立了复杂性结果并设计了一个基于约束多线性筛选的代数算法框架。我们证明了我们的解决方案可以扩展到具有多达 10 亿条边的海量图（对于具有五种颜色的多集查询）和多达 10 亿条边（对于具有 10 种颜色的多集查询），尽管问题是 NP 难的。我们的实施，它是公开可用的，展示了实用的边缘线性可扩展性并且高度优化。例如，在具有超过 600 万条边和具有十种颜色的多集查询的真实世界图形数据集中，我们可以在具有四核的 Haswell 桌面上在不到八分钟的时间内提取出最佳解决方案。