# An Algorithm for Solving the Shortest Path Improvement Problem on Rooted Trees Under Unit Hamming Distance

- 121 Downloads

## Abstract

Shortest path problems play important roles in computer science, communication networks, and transportation networks. In a shortest path improvement problem under unit Hamming distance, an edge-weighted graph with a set of source–terminal pairs is given. The objective is to modify the weights of the edges at a minimum cost under unit Hamming distance such that the modified distances of the shortest paths between some given sources and terminals are upper bounded by the given values. As the shortest path improvement problem is NP-hard, it is meaningful to analyze the complexity of the shortest path improvement problem defined on rooted trees with one common source. We first present a preprocessing algorithm to normalize the problem. We then present the proofs of some properties of the optimal solutions to the problem. A dynamic programming algorithm is proposed for the problem, and its time complexity is analyzed. A comparison of the computational experiments of the dynamic programming algorithm and MATLAB functions shows that the algorithm is efficient although its worst-case complexity is exponential time.

## Keywords

Shortest path problem Rooted trees Network improvement problem Hamming distance Dynamic programming## Mathematics Subject Classification

90C27 90C35 90C39## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (11471073) and Chinese Universities Scientific Fund (2015B27914). The work of P.M. Pardalos was conducted at the National Research University Higher School of Economics and supported by the RSF Grant 14-41-00039.

## References

- 1.Heuberger, C.: Inverse optimization: a survey on problems, methods, and results. J. Comb. Optim.
**8**, 329–361 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Burton, D., Toint, P.: On an instance of the inverse shortest path problem. Math. Program.
**53**, 45–61 (1992)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Zhang, J., Ma, Z., Yang, C.: A column generation method for inverse shortest path problems. ZOR Math. Method Oper. Res.
**41**, 347–358 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Zhang, J., Lin, Y.: Computation of the reverse shortest-path problem. J. Global Optim.
**25**, 243–261 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 5.He, Y., Zhang, B., Yao, E.: Weighted inverse minimum spanning tree problems under Hamming distance. J. Comb. Optim.
**9**(1), 91–100 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Volgenant, D.C.: A some inverse optimization problems under the Hamming distance. Eur. J. Oper. Res.
**170**, 887–899 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Guan, X.C., Zhang, B.W.: Inverse 1-median problem on trees under weighted Hamming distance. J. Global Optim.
**54**(1), 75–82 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Guan, X.C., He, X.Y., Pardalos, P.M., Zhang, B.W.: Inverse max+sum spanning tree problem under hamming distance by modifying the sum-cost vector. J. Global Optim.
**69**(4), 911–925 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Jiang, Y., Liu, L., Wu, B., Yao, E.: Inverse minimum cost flow problems under the weighted Hamming distance. Eur. J. Oper. Res.
**207**, 50–54 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Liu, L., Yao, E.: Inverse min–max spanning tree problem under the weighted sum-type Hamming distance. Theor. Comput. Sci.
**396**, 28–34 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Zhang, B., Zhang, J., He, Y.: The center location improvement under Hamming distance. J. Comb. Optim.
**9**(2), 187–198 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Zhang, B., Zhang, J., He, Y.: Constrained inverse minimum spanning tree problems under bottleneck-type Hamming distance. J. Global Optim.
**34**, 467–474 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Zhang, B., Zhang, J., Qi, L.: The shortest path improvement problem under Hamming distance. J. Comb. Optim.
**12**(4), 351–361 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Zhang, B., Guan, X., He, C., Wang, S.: Algorithms for the shortest path improvement problems under unit Hamming distance. J. Appl. Math. (2013). https://doi.org/10.1155/2013/847317 MathSciNetGoogle Scholar
- 15.Zhang, B., Guan, X., Wang, Q., He, C., Samson, H.: The complexity analysis of the shortest path improvement problem under the Hamming distance. Pac. J. Optim.
**11**(4), 605–608 (2015)MathSciNetzbMATHGoogle Scholar