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arXiv · 2404.10638

A Fast 3-Approximation for the Capacitated Tree Cover Problem with Edge Loads

Benjamin Rockel-Wolff
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Abstract

The capacitated tree cover problem with edge loads is a variant of the tree cover problem, where we are given facility opening costs, edge costs and loads, as well as vertex loads. We try to find a tree cover of minimum cost such that the total edge and vertex load of each tree does not exceed a given bound. We present an $\mathcal{O}(m\log n)$ time 3-approximation algorithm for this problem. This is achieved by starting with a certain LP formulation. We give a combinatorial algorithm that solves the LP optimally in time $\mathcal{O}(m\log n)$. Then, we show that a linear time rounding and splitting technique leads to an integral solution that costs at most 3 times as much as the LP solution. Finally, we prove that the integrality gap of the LP is $3$, which shows that we can not improve the rounding step in general.

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Benjamin Rockel-Wolff. 2024-04-16. A Fast 3-Approximation for the Capacitated Tree Cover Problem with Edge Loads. https://arxiv.org/abs/2404.10638

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