Abstract
Given a 2-edge-connected, real weighted graph G with n vertices and m edges, the 2-edge-connectivity augmentation problem is that of finding a minimum weight set of edges of G to be added to a spanning subgraph H of G to make it 2-edge-connected. While the general problem is NP-hard and 2 -approximable, in this paper we prove that it becomes polynomial time solvable if H is a depth-first search tree of G . More precisely, we provide an efficient algorithm for solving this special case which runs in O(M · α(M,n)) time, where α is the classic inverse of Ackermann's function and M=m · α(m,n) . This algorithm has two main consequences: first, it provides a faster 2 -approximation algorithm for the general 2 -edge-connectivity augmentation problem; second, it solves in O(m · α(m,n)) time the problem of restoring, by means of a minimum weight set of replacement edges, the 2 -edge-connectivity of a 2-edge-connected communication network undergoing a link failure.
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Galluccio, A., Proietti, G. Polynomial Time Algorithms for 2-Edge-Connectivity Augmentation Problems . Algorithmica 36, 361–374 (2003). https://doi.org/10.1007/s00453-003-1024-7
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DOI: https://doi.org/10.1007/s00453-003-1024-7