Abstract
We consider a special case of the generalized minimum spanning tree problem (GMST) and the generalized travelling salesman problem (GTSP) where we are given a set of points inside the integer grid (in Euclidean plane) where each grid cell is \(1 \times 1\). In the MST version of the problem, the goal is to find a minimum tree that contains exactly one point from each non-empty grid cell (cluster). Similarly, in the TSP version of the problem, the goal is to find a minimum weight cycle containing one point from each non-empty grid cell. We give a \((1+4\sqrt{2}+\epsilon )\) and \((1.5+8\sqrt{2}+\epsilon )\)-approximation algorithm for these two problems in the described setting, respectively.
Our motivation is based on the problem posed in [6] for a constant approximation algorithm. The authors designed a PTAS for the more special case of the GMST where non-empty cells are connected end dense enough. However, their algorithm heavily relies on this connectivity restriction and is unpractical. Our results develop the topic further.
Supported by NSERC Canada.
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We would like to thank Geoffrey Exoo for many usefull discussions.
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Bhattacharya, B., Ćustić, A., Rafiey, A., Rafiey, A., Sokol, V. (2015). Approximation Algorithms for Generalized MST and TSP in Grid Clusters. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_9
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