 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.55
URN: urn:nbn:de:0030-drops-126581
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12658/
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### A Constant Factor Approximation for Capacitated Min-Max Tree Cover

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### Abstract

Given a graph G = (V,E) with non-negative real edge lengths and an integer parameter k, the (uncapacitated) Min-Max Tree Cover problem seeks to find a set of at most k trees which together span V and each tree is a subgraph of G. The objective is to minimize the maximum length among all the trees. In this paper, we consider a capacitated generalization of the above and give the first constant factor approximation algorithm. In the capacitated version, there is a hard uniform capacity (λ) on the number of vertices a tree can cover. Our result extends to the rooted version of the problem, where we are given a set of k root vertices, R and each of the covering trees is required to include a distinct vertex in R as the root. Prior to our work, the only result known was a (2k-1)-approximation algorithm for the special case when the total number of vertices in the graph is kλ [Guttmann-Beck and Hassin, J. of Algorithms, 1997]. Our technique circumvents the difficulty of using the minimum spanning tree of the graph as a lower bound, which is standard for the uncapacitated version of the problem [Even et al.,OR Letters 2004] [Khani et al.,Algorithmica 2010]. Instead, we use Steiner trees that cover λ vertices along with an iterative refinement procedure that ensures that the output trees have low cost and the vertices are well distributed among the trees.

### BibTeX - Entry

```@InProceedings{das_et_al:LIPIcs:2020:12658,
author =	{Syamantak Das and Lavina Jain and Nikhil Kumar},
title =	{{A Constant Factor Approximation for Capacitated Min-Max Tree Cover}},
booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
pages =	{55:1--55:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-164-1},
ISSN =	{1868-8969},
year =	{2020},
volume =	{176},
editor =	{Jaros{\l}aw Byrka and Raghu Meka},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12658},
URN =		{urn:nbn:de:0030-drops-126581},
doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.55},
annote =	{Keywords: Approximation Algorithms, Graph Algorithms, Min-Max Tree Cover, Vehicle Routing, Steiner Tree}
}
```

 Keywords: Approximation Algorithms, Graph Algorithms, Min-Max Tree Cover, Vehicle Routing, Steiner Tree Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020) Issue Date: 2020 Date of publication: 11.08.2020

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