# A Hamiltonian Approach to the Assignment of Non-reusable Frequencies

## Abstract

The problem of Radio Labelling is to assign distinct integer labels to all the vertices of a graph, such that adjacent vertices get labels at distance at least two. The objective is to minimize the label span. Radio labelling is a combinatorial model for frequency assignment in case that the transmitters are not allowed to operate at the same channel.

We show that radio labelling is related to TSP(1,2). Hence, it is \({\cal NP}\)-complete and MAX-SNP-hard. Then, we present a polynomial-time algorithm for computing an optimal radio labelling, given a coloring of the graph with constant number of colors. Thus, we prove that radio labelling is in \({\cal P}\) for planar graphs. We also obtain a \(\frac{3}{2}\)-approximation \({\cal NC}\) algorithm and we prove that approximating radio labelling in graphs of bounded maximum degree is essentially as hard as in general graphs.

We obtain similar results for TSP(1,2). In particular, we present the first \(\frac{3}{2}\)-approximation \({\cal NC}\) algorithm for TSP(1,2), and we prove that dense instances of TSP(1,2) do not admit a PTAS, unless \({\cal P}= {\cal NP}\).

## Keywords

Planar Graph Hamiltonian Cycle Hamiltonian Path Maximal Match Polynomial Time Approximation Scheme## Preview

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## References

- 1.Allen, S.M., Smith, D.H., Hurley, S.: Lower Bounding Techniques for Frequency Assignment. Submitted to Discrete Mathematics (1997)Google Scholar
- 2.Christofides, N.: Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem. In: Symposium on new directions and recent results in algorithms and complexity, p. 441 (1976)Google Scholar
- 3.Fernandez de la Vega, W., Karpinski, M.: On Approximation Hardness of Dense TSP and other Path Problems. Electronic Colloquium on Computational Complexity TR98-024 (1998)Google Scholar
- 4.Fotakis, D., Pantziou, G., Pentaris, G., Spirakis, P.: Frequency Assignment in Mobile and Radio Networks. To appear in Proc. of the Workshop on Networks in Distributed Computing. DIMACS Series, AMS, Providence (1998)Google Scholar
- 5.Gower, R.A.H., Leese, R.A.: The Sensitivity of Channel Assignment to Constraint Specification. In: Proc. of the 12th International Symposium on Electromagnetic Compatibility, pp. 131–136 (1997)Google Scholar
- 6.Hale, W.K.: Frequency Assignment: Theory and Applications. Proceedings of the IEEE 68(12), 1497–1514 (1980)CrossRefGoogle Scholar
- 7.Harary, F.: Personal Communication (1997)Google Scholar
- 8.van den Heuvel, J., Leese, R.A., Shepherd, M.A.: Graph Labelling and Radio Channel Assignment (1996), Manuscript available from http://www.maths.ox.ac.uk/users/gowerr/preprints.html
- 9.Israeli, A., Shiloach, Y.: An improved algorithm for maximal matching. Information Processing Letters 33, 57–60 (1986)CrossRefMathSciNetGoogle Scholar
- 10.Karpinski, M.: Polynomial Time Approximation Schemes for Some Dense Instances of
*NP*-hard Optimization Problems. In: Proc. of the 1st Symposium on Randomization and Approximation Techniques in Computer Science, pp. 1–14 (1997)Google Scholar - 11.Khanna, S., Kumaran, K.: On Wireless Spectrum Estimation and Generalized Graph Coloring. In: Proc. of the 17th Joint Conference of IEEE Computer and Communications Societies - INFOCOM (1998)Google Scholar
- 12.Papadimitriou, C.H.: Personal Communication (1998)Google Scholar
- 13.Papadimitriou, C.H., Yannakakis, M.: The Traveling Salesman Problem with Distances One and Two. Mathematics of Operations Research 18(1), 1–11 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
- 14.Raychaudhuri, A.: Intersection assignments, T-colourings and powers of graphs. PhD Thesis, Rutgers University (1985)Google Scholar