The paper focuses on the problem of finding the most degree-central shortest path in a graph. The authors first propose a polynomial-time algorithm (Algorithm 1) to solve this problem for unweighted graphs. The algorithm is based on a modified breadth-first search (BFS) approach and has a worst-case running time of O(|E||V|²∆(G)), where |V| is the number of vertices, |E| is the number of edges, and ∆(G) is the maximum degree of the graph.
The authors then show that the problem becomes NP-hard when the graph is weighted, even with just two distinct weights. They discuss two special cases for weighted graphs: one with positive integer weights and one with weights drawn from a continuous distribution. For the former case, they propose a modified version of Algorithm 1 that can solve the problem in pseudo-polynomial time. For the latter case, they show that the problem can be solved in polynomial time by first finding all-pairs shortest paths and then evaluating the centralities.
The paper also considers two other centrality measures - betweenness centrality and closeness centrality. For the betweenness-central shortest path problem, the authors show that it can be solved in polynomial time for both weighted and unweighted graphs. However, they prove that the problem of finding the most closeness-central shortest path is NP-hard, regardless of whether the graph is weighted or not.
The authors conduct extensive computational experiments on both synthetic and real-world graph instances to compare the performance of their proposed algorithm against the existing MVP algorithm. The results demonstrate significant improvements in runtime, especially for larger graph instances.
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by Johnson Phos... pada arxiv.org 05-07-2024
https://arxiv.org/pdf/2401.08019.pdfPertanyaan yang Lebih Dalam