Approximation Algorithms for Generalized MST and TSP in Grid Clusters [article]

Binay Bhattacharya, Ante Ćustić, Akbar Rafiey, Arash Rafiey, Vladyslav Sokol
2015 arXiv   pre-print
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 × 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
more » ... t from each non-empty grid cell. We give a (1+4√(2)+ϵ) and (1.5+8√(2)+ϵ)-approximation algorithm for these two problems in the described setting, respectively. Our motivation is based on the problem posed in [7] 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.
arXiv:1507.04438v1 fatcat:bu3wz75myfevblogdzy2rs77b4