A 5/4-Approximation Algorithm for Biconnecting a Graph with a Given Hamiltonian Path

Finding a minimum size 2-vertex connected spanning subgraph of a k-vertex connected graph G = (V,E) with n vertices and m edges is known to be NP-hard and APX-hard, as well as approximable in O(n 2 m) time within a factor of 4/3. Interestingly, the problem remains NP-hard even if a Hamiltonian path of G is given as part of the input. For this input-enriched version of the problem, we provide in this paper a linear time and space algorithm which approximates the optimal solution by a factor of no more than min {54,2k−12(k−1)} .