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On the VC-Dimension of Unique Round-Trip Shortest Path Systems.
Information Processing Letters ( IF 0.5 ) Pub Date : 2019-01-10 , DOI: 10.1016/j.ipl.2019.01.001
Chun Jiang Zhu 1 , Kam-Yiu Lam 2 , Joseph Kee Yin Ng 3 , Jinbo Bi 1
Affiliation  

The VC-dimension, which has wide uses in learning theory, has been used in the analysis and design of graph algorithms recently. In this paper, we study the problem of bounding the VC-dimension of unique round-trip shortest path set systems (URTSP), which are set systems induced by sets of vertices in unique round-trip shortest paths in directed graphs. We first show that different from the VC-dimensions of set systems induced by unique undirected and directed shortest paths in undirected and directed graphs respectively, the VC-dimension of URTSP can be larger than 3. We then prove that the VC-dimension of URTSP is at most 32. Furthermore, we apply the VC-dimension result to the minimum k-round-trip shortest path cover problem (k-RTSPC), which is to find for a directed graph a minimum vertex set to intersect every round-trip shortest path containing at least k vertices, and derive an upper bound on the size of the vertex set. The k-RTSPC problem can be useful in many real-world applications, including optimal placement of facilities.



中文翻译:

关于唯一往返最短路径系统的VC维。

VC维度在学习理论中得到了广泛的应用,最近已被用于图算法的分析和设计中。在本文中,我们研究了限制唯一双向最短路径集系统(URTSP)的VC维问题的方法,该系统是由有向图中唯一双向最短路径的顶点集诱发的集合系统。我们首先表明,不同于分别由无向和有向图中唯一的无向和有向最短路径引起的集合系统的VC维,URTSP的VC维可以大于3。然后证明了URTSP的VC维。最多为32。此外,我们将VC维结果应用于最小的k往返最短路径覆盖问题(ķ -[RŤ小号PC),这将为有向图找到一个最小的顶点集,以与每个至少包含k个顶点的往返最短路径相交,并得出顶点集大小的上限。该ķ -[RŤ小号PC 该问题在许多实际应用中都可能有用,包括设施的最佳放置。

更新日期:2019-01-10
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