Abstract
Given a graph G = (V,E) with nonnegative real edge lengths and an integer parameter k, the (uncapacitated) MinMax 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 (2k1)approximation algorithm for the special case when the total number of vertices in the graph is kλ [GuttmannBeck 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 MinMax Tree Cover}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
pages = {55:155:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771641},
ISSN = {18688969},
year = {2020},
volume = {176},
editor = {Jaros{\l}aw Byrka and Raghu Meka},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12658},
URN = {urn:nbn:de:0030drops126581},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.55},
annote = {Keywords: Approximation Algorithms, Graph Algorithms, MinMax Tree Cover, Vehicle Routing, Steiner Tree}
}
Keywords: 

Approximation Algorithms, Graph Algorithms, MinMax 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 