The approximability of the weighted Hamiltonian path completion problem on a tree

Given a graph, the Hamiltonian path completion problem is to find an augmenting edge set such that the augmented graph has a Hamiltonian path. In this paper, we show that the Hamiltonian path completion problem will unlikely have any constant ratio approximation algorithm unless NP = P. This problem remains hard to approximate even when the given subgraph is a tree. Moreover, if the edge weights are restricted to be either 1 or 2, the Hamiltonian path completion problem on a tree is still NP-hard. Then it is observed that this problem is strongly NP-hard, so it does not have any fully polynomial-time approximation scheme (FPTAS) unless NP=P. When the given tree is a k-tree, we give an approximation algorithm with performance ratio 1.5.