# Problem of Determining Weights of Edges for Reducing Diameter

## Abstract

It is necessary to reduce delay in an information network. The total delay time between two nodes is the sum of the delay times in all links and nodes contained in the path between the nodes; therefore, we can reduce the total delay time by reducing the delay times in some links and nodes in the path. Since the additional cost such as the increase of the link speed is necessary, the links whose speed is increased must be determined by considering the trade-off between performance and cost. In this paper, we formulate this network design problem as an optimization problem of determining the weight of each edge so that the diameter of a network is less than or equal to a given threshold. The objective of this optimization problem is to minimize the sum of the costs under the condition that the number of edges whose weight is changed is restricted. We prove that this problem is NP-complete, and we propose a polynomial-time algorithm to the problem that the number of edges whose weights are changed is restricted to one.

## Notes

### Acknowledgements

This work was partially supported by the Japan Society for the Promotion of Science through Grants-in-Aid for Scientific Research (B) (17H01742) and JST CREST JPMJCR1402.

## References

- 1.de Bruijn, N.G.: A combinatorial problem. In: Proceedings of Koninklijke Nederlandse Academie van Wetenschappen, vol. 49, pp. 758–764 (1946)Google Scholar
- 2.Kautz, W.: Bounds on directed (d,k)-graphs. In: Theory of Cellular Logic Networks and Machines (1968)Google Scholar
- 3.Du, D.Z., Gao, F., Hsu, D.F.: De Bruiin digraphs, Kautz digraphs, and their generalizations. In: Combinatorial Network Theory, vol. 1, pp. 65–105 (1996)CrossRefGoogle Scholar
- 4.Boucher, C., Bowe, A., Gagie, T., Puglisi, S.J., Sadakane, K.: Variable-order de Bruijn graphs. In: Proceedings of IEEE Data Compression Conference, 7–9 April 2015Google Scholar
- 5.Bermond, J.-C., Peyrat, C.: De Bruijn and Kautz networks: a competitor for the hypercube? In: Hypercube and Distributed Computers, pp. 279–293 (1989)Google Scholar
- 6.Imase, M., Itoh, M.: A design for directed graphs with minimum diameter. IEEE Trans. Comput.
**C–32**(8), 782–784 (1983)CrossRefGoogle Scholar - 7.Du, D.Z., Hsu, D.F., Peck, G.W.: Connectivity of consecutive-d digraph. Discret. Appl. Math.
**37/38**, 169–177 (1992)MathSciNetCrossRefGoogle Scholar - 8.Plesnik, J.: The complexity of designing a network with minimum diameter. Networks
**11**(1), 77–85 (1981)MathSciNetCrossRefGoogle Scholar - 9.Miwa, H., Ito, H.: Sparse spanning subgraph preserving connectivity and distance between vertices and vertex subsets. IEICE Trans. Fundam.
**E81–A**(5), 832–841 (1998)Google Scholar - 10.Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness (1978)Google Scholar
- 11.CAIDA. http://www.caida.org/