A PTAS for k-hop MST on the Euclidean plane: Improving Dependency on k

06/21/2021

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by   Jittat Fakcharoenphol, et al.

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For any ϵ>0, Laue and Matijević [CCCG'07, IPL'08] give a PTAS for finding a (1+ϵ)-approximate solution to the k-hop MST problem in the Euclidean plane that runs in time (n/ϵ)^O(k/ϵ). In this paper, we present an algorithm that runs in time (n/ϵ)^O(log k ·(1/ϵ)^2·log^2(1/ϵ)). This gives an improvement on the dependency on k on the exponent, while having a worse dependency on ϵ. As in Laue and Matijević, we follow the framework introduced by Arora for Euclidean TSP. Our key ingredients include exponential distance scaling and compression of dynamic programming state tables.