Abstract
In this paper we study the hardness of the kCenter problem on inputs that model transportation networks. For the problem, an edgeweighted graph G=(V,E) and an integer k are given and a center set C subseteq V needs to be chosen such that C<= k. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist. We complement these results by proving that the kCenter problem is W[1]hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers k, the highway dimension h, and even the treewidth t. Moreover, under the Exponential Time Hypothesis there is no f(k,t,h)* n^{o(t+sqrt{k+h})} time algorithm for any computable function f. Thus it is unlikely that the optimum solution to kCenter can be found efficiently, even when assuming that the input graph abides to all of the above models for transportation networks at once!
Additionally we give a simple parameterized (1+{epsilon})approximation algorithm for inputs of doubling dimension d with runtime (k^k/{epsilon}^{O(kd)})* n^{O(1)}. This generalizes a previous result, which considered inputs in Ddimensional L_q metrics.
BibTeX  Entry
@InProceedings{feldmann_et_al:LIPIcs:2018:8845,
author = {Andreas Emil Feldmann and D{\'a}niel Marx},
title = {{The Parameterized Hardness of the kCenter Problem in Transportation Networks}},
booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)},
pages = {19:119:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770682},
ISSN = {18688969},
year = {2018},
volume = {101},
editor = {David Eppstein},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8845},
URN = {urn:nbn:de:0030drops88450},
doi = {10.4230/LIPIcs.SWAT.2018.19},
annote = {Keywords: kcenter, parameterized complexity, planar graphs, doubling dimension, highway dimension, treewidth}
}
Keywords: 

kcenter, parameterized complexity, planar graphs, doubling dimension, highway dimension, treewidth 
Collection: 

16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018) 
Issue Date: 

2018 
Date of publication: 

04.06.2018 