Local PTAS for Independent Set and Vertex Cover in Location Aware Unit Disk Graphs
We present the first local approximation schemes for maximum independent set and minimum vertex cover in unit disk graphs. In the graph model we assume that each node knows its geographic coordinates in the plane (location aware nodes). Our algorithms are local in the sense that the status of each node v (whether or not v is in the computed set) depends only on the vertices which are a constant number of hops away from v. This constant is independent of the size of the network. We give upper bounds for the constant depending on the desired approximation ratio. We show that the processing time which is necessary in order to compute the status of a single vertex is bounded by a polynomial in the number of vertices which are at most a constant number of vertices away from it. Our algorithms give the best possible approximation ratios for this setting.
The technique which we use to obtain the algorithm for vertex cover can also be employed for constructing the first global PTAS for this problem in unit disk graph which does not need the embedding of the graph as part of the input.
KeywordsApproximation Ratio Local Algorithm Class Number Vertex Cover Unit Disk Graph
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- 5.Czyzowicz, J., Dobrev, S., Fevens, T., González-Aguilar, H., Kranakis, E., Opatrny, J., Urrutia, J.: Local algorithms for dominating and connected dominating sets of unit disc graphs with location aware nodes. In: Proceedings of LATIN 2008. LNCS, vol. 4957 (2008)Google Scholar
- 7.Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness (1979)Google Scholar
- 8.Håstad, J.: Clique is hard to approximate within n 1 − ε. Electronic Colloquium on Computational Complexity (ECCC) 4(38) (1997)Google Scholar
- 14.Nieberg, T., Hurink, J.L., Kern, W.: A robust PTAS for maximum weight independent sets in unit disk graphs. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 214–221. Springer, Heidelberg (2004)Google Scholar
- 15.Wiese, A., Kranakis, E.: Local PTAS for Dominating and Connected Dominating Set in Location Aware UDGs (to appear, 2007)Google Scholar