An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree

Given a graph G=(V,E) and a tree T =(V,F) with E ∩ F = φ such that G + T =(V,F ∪ E) is 2-edge-connected, we consider the problem of finding a smallest 2-edge-connected spanning subgraph (V,F ∪ E') of G + T containing T. The problem, which is known to be NP-hard, admits a 2-approximation algorithm. However, obtaining a factor better than 2 for this problem has been one of the main open problems in the graph augmentation problem. In this paper, we show that the problem is (1.875 + e)-approximable in O(n½m + n2) time for any constant e < 0, where n = |V| and m = |E ∪ F|.

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