Approximation Algorithm for the Minimum Hub Cover Set Problem

• Published in: IEEE access Vol. 10; pp. 51419 - 51427
• Main Authors: Trejo-Sanchez, Joel A., Sansores-Perez, Candelaria E., Garcia-Diaz, Jesus, Fernandez-Zepeda, Jose Alberto
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Apexes; Approximation; Approximation algorithms; Arrays; Graph theory; Graphs; Heuristic algorithms; heuristics; Mathematical analysis; minimum hub cover set; NP-hard problem; optimization; Satellites; STEM
• ISSN: 2169-3536; 2169-3536
• DOI: 10.1109/ACCESS.2022.3173615

A subset <inline-formula> <tex-math notation="LaTeX">{\mathcal{ S}}\subseteq V </tex-math></inline-formula> of vertices of an undirected graph <inline-formula> <tex-math notation="LaTeX">G=(V,E) </tex-math></inline-formula> is a hub cover when for each edge <inline-formula> <tex-math notation="LaTeX">(u,v) \in E </tex-math></inline-formula>, at least one of its endpoints belongs to <inline-formula> <tex-math notation="LaTeX">{\mathcal{ S}} </tex-math></inline-formula>, or there exists a vertex <inline-formula> <tex-math notation="LaTeX">r \in {\mathcal{ S}} </tex-math></inline-formula> that is a neighbor of both <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula>. The problem of computing a minimum hub cover set in arbitrary graphs is NP-hard. This problem has applications for indexing large databases. This paper proposes <inline-formula> <tex-math notation="LaTeX">\Psi </tex-math></inline-formula>-MHC, the first approximation algorithm for the minimum hub cover set in arbitrary graphs to the best of our knowledge. The approximation ratio of this algorithm is <inline-formula> <tex-math notation="LaTeX">\ln \mu </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula> is upper bounded by <inline-formula> <tex-math notation="LaTeX">\min \left\{{\frac {1}{2}(\Delta +1)^{2}, \vert E\vert }\right\} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\Delta </tex-math></inline-formula> is the degree of <inline-formula> <tex-math notation="LaTeX">G </tex-math></inline-formula>. The execution time of <inline-formula> <tex-math notation="LaTeX">\Psi </tex-math></inline-formula>-MHC is <inline-formula> <tex-math notation="LaTeX">O((\Delta + 1) \vert E \vert + \vert {\mathcal{ S}} \vert \vert V \vert) </tex-math></inline-formula>. Experimental results show that <inline-formula> <tex-math notation="LaTeX">\Psi </tex-math></inline-formula>-MHC far outperforms the theoretical approximation ratio for the input graph instances.