A 10/7-Approximation for Discrete Bamboo Garden Trimming and Continuous Trimming on Star Graphs

In the discrete bamboo garden trimming problem we are given n bamboo that grow at rates v1, . . ., vn per day. Each day a robotic gardener cuts down one bamboo to height 0. The goal is to find a schedule that minimizes the height of the tallest bamboo that ever exists. We present a 10/7-approximation algorithm that is based on a reduction to the pinwheel problem. This is consistent with the approach of earlier algorithms, but some new techniques are used that lead to a better approximation ratio. We also consider the continuous version of the problem where the gardener travels in a metric space between plants and cuts down a plant each time he reaches one. We show that on the star graph the previously proposed algorithm Reduce-Fastest is a 6-approximation and the known Deadline-Driven Strategy is a (3 + 2√2)-approximation. The Deadline-Driven Strategy is also a (9 + 2√5)-approximation on star graphs with multiple plants on each branch.