A Faster Approximation Algorithm for 2-Edge-Connectivity Augmentation

Abstract

Given a 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. Such a problem is well-known to be NP-hard, but it becomes solvable in polynomial time if H is a depth-first search tree of G, and the fastest algorithm for this special case runs in $\mathcal{O}\left( {m + n\log n} \right)$ time. In this paper, we sensibly improve such a bound, by providing an efficient algorithm running in $\mathcal{O}\left( {M \cdot \alpha \left( {M,n} \right)} \right)$ time, where α is the classic inverse of the 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 $\mathcal{O}\left( {M \cdot \alpha \left( {M,n} \right)} \right)$ time the problem of maintaining, by means of a minimum weight set of edges, the 2-edge-connectivity of a 2-edge-connected communication network undergoing an edge failure, thus improving the previous $\mathcal{O}\left( {m + n\log n} \right)$ time bound.

This work has been partially supported by the CNR-Agenzia 2000 Program, under Grants No. CNRC00CAB8 and CNRG003EF8, and by the Research Project REAL-WINE, partially funded by the Italian Ministry of Education, University and Research.