Abstract
We introduce a new framework of Airport and Railway Problems, which combines capacitated facility location with network design. In this framework we are given a graph with weights on the vertices and on the edges, together with a parameter k. The vertices of the graph represent cities, and weights denote respectively the costs of opening airports in the cities and building railways that connect pairs of cities. The parameter $k$ can be thought of as the capacity of an airport. The goal is to construct a minimum cost network of airports and railways connecting the cities, where each connected component in the network spans at most k vertices, contains an open airport, and the network satisfies some additional requirements specific to the problem in the framework.
We consider two problems in this framework. In the AR_F problem there are no additional requirements for the network. This problem is related to capacitated facility location. In the AR_P problem, we require each component to be a path with airports at both endpoints. AR_P is a relaxation of the capacitated vehicle routing problem (CVRP).
We consider the problems in the twodimensional Euclidean setting. We show that both AR_F and AR_P are NPhard, even for uniform vertex weights (i.e., when the cost of building an airport is the same for all cities).
On the positive side, we provide polynomial time approximation schemes for AR_F and AR_P when vertex weights are uniform. We also investigate AR_F and AR_P for k = infinity. In this setting we present an exact polynomial time algorithm for AR_F with general vertex costs, which also works for general edge costs. In contrast to AR_F, AR_P remains NPhard when k = infinity, and we present a polynomial time approximation scheme for general vertex weights.
We believe that our PTAS for AR_P with uniform vertex weights and arbitrary k brings us closer towards a PTAS for Euclidean CVRP, for which the main difficulty is to deal with paths of length at most k.
BibTeX  Entry
@InProceedings{adamaszek_et_al:LIPIcs:2016:5707,
author = {Anna Adamaszek and Antonios Antoniadis and Tobias M{\"o}mke},
title = {{Airports and Railways: Facility Location Meets Network Design}},
booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
pages = {6:16:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770019},
ISSN = {18688969},
year = {2016},
volume = {47},
editor = {Nicolas Ollinger and Heribert Vollmer},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5707},
URN = {urn:nbn:de:0030drops57074},
doi = {10.4230/LIPIcs.STACS.2016.6},
annote = {Keywords: approximation algorithms, geometric approximation, facility location, network design, PTAS}
}
Keywords: 

approximation algorithms, geometric approximation, facility location, network design, PTAS 
Collection: 

33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016) 
Issue Date: 

2016 
Date of publication: 

16.02.2016 