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

Abstract

Given a graph G=( V, E) and a tree T=( V, F) with E∩ F=∅ 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+ ε)-approximable in O( n 1/2 m+ n 2) time for any constant ε>0, where n=| V| and m=| E∪ F|.