NP-Completeness Results and Efficient Approximations for Radiocoloring in Planar Graphs
The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. The Radiocoloring (RC) of a graph G(V,E) is an assignment function Φ: V → IN such that ¦Φ(u)-Φ(v)≥ 2, when u; v are neighbors in G, and ¦Φ(u)-Φ(v)≥1 when the minimum distance of u; v in G is two. The discrete number and the range of frequencies used are called order and span, respectively. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span or the order. In this paper we prove that the min span RCP is NP-complete for planar graphs. Next, we provide an O(nΔ) time algorithm (¦V¦ = n) which obtains a radiocoloring of a planar graph G that approximates the minimum order within a ratio which tends to 2 (where Δ the maximum degree of G). Finally, we provide a fully polynomial randomized approximation scheme (fpras) for the number of valid radiocolorings of a planar graph G with λ colors, in the case λ ≥ 4λ + 50.
KeywordsPlanar Graph Chromatic Number Minimum Order Optimization Version Central Vertex
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- 1.D. Aldous: Random walks in finite groups and rapidly mixing Markov Chains. Seminaire de Probabilites XVII 1981/82 (A. Dold and B. Eckmann, eds), Springer Lecture Notes in Mathematis986 (1982) 243–297.Google Scholar
- 2.Geir Agnarsson, Magnus M. Hallorsson: Coloring Powers of planar graphs. Symposium of Discrete Algorithms (2000).Google Scholar
- 5.D. Fotakis, G. Pantziou, G. Pentaris and P. Spirakis: Frequency Assignment in Mobile and Radio Networks. Networks in Distributed Computing, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 45 American Mathematical Society (1999) 73–90.Google Scholar
- 6.D.A. Fotakis, S.E. Nikoletseas, V.G. Papadopoulou and P.G. Spirakis: NP-completeness Results and Efficient Approximations for Radiocoloring in Planar Graphs. CTI Technical Report (2000) (url: http://www.cti.gr/RD1).
- 7.D. Fotakis and P. Spirakis: Assignment of Reusable and Non-Reusable Frequencies. International Conference on Combinatorial and Global Optimization (1998).Google Scholar
- 8.J. Van D. Heuvel and S. McGuiness: Colouring the square of a Planar Graph. CDAM Research Report Series, Jule (1999).Google Scholar
- 10.M. Jerrum: Markov Chain Monte Carlo Method. Probabilistic Methods for Algorithmic Discrete Mathematics, Springer (1998).Google Scholar
- 11.S. Ramanathan, E. R. Loyd: Scheduling algorithms for Multi-hop Radio Networks. IEEE/ACM Trans. on Networking, 1(2): (April) 1993) 166–172.Google Scholar
- 12.S. Ramanathan, E. R. Loyd: The complexity of distance2-coloring. 4th International Conference of Computing and information, (1992) 71–74.Google Scholar