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

  • Authors:

  • Quincy Wu;Chin Lung Lu;Richard Chia-Tung Lee

  • Affiliations:

  • Graduate Institute of Communication Engineering, National Chi-Nan University, Puli, Taiwan, ROC;Department of Biological Science and Technology, National Chiao Tung University, Hsinchu, Taiwan, ROC;Department of Computer Science and Information Engineering, National Chi-Nan University, Puli, Taiwan, ROC

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2005

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Abstract

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.