On the VC-dimension of unique round-trip shortest path systems

Abstract

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 VCdimension 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.

Parallel Machine Learning Algorithm Using Fine-Grained-Mode Spark on a Mesos Big Data Cloud Computing Software Framework for Mobile Robotic Intelligent Fault Recognition

“…SUPPORT VECTOR MACHINE LEARNING FOR ROLLER BEARING FAULT DIAGNOSIS PROCESSA. SUPPORT VECTOR MACHIMECorinna Cortes and Vapnik developed the currently used SVM technique to minimize the VC dimension[27][28][29][30][31]. The hyperplane, which separates data (…”