Approximating Airports and Railways
In this paper we consider the airport and railway problem (AR), which combines capacitated facility location with network design, both in the general metric and the two-dimensional Euclidean space. An instance of the airport and railway problem consists of a set of points in the corresponding metric, together with a non-negative weight for each point, and a parameter k. The points represent cities, the weights denote costs of opening an airport in the corresponding city, and the parameter k is a maximum capacity of an airport. The goal is to construct a minimum cost network of airports and railways connecting all the cities, where railways correspond to edges connecting pairs of points, and the cost of a railway is equal to the distance between the corresponding points. The network is partitioned into components, where each component contains an open airport, and spans at most k cities. For the Euclidean case, any points in the plane can be used as Steiner vertices of the network. We obtain the first bicriteria approximation algorithm for AR for the general metric case, which yields a 4-approximate solution with a resource augmentation of the airport capacity k by a factor of 2. More generally, for any parameter 0 < p <= 1 where pk is an integer we develop a (4/3)(2 + 1/p)-approximation algorithm for metric AR with a resource augmentation by a factor of 1 + p.
Furthermore, we obtain the first constant factor approximation algorithm that does not resort to resource augmentation for AR in the Euclidean plane. Additionally, for the Euclidean setting we provide a quasi-polynomial time approximation scheme for the same problem with a resource augmentation by a factor of 1 + mu on the airport capacity, for any fixed mu > 0.
Network Design
Facility Location
Approximation Algorithms
PTAS
Metric
Euclidean
5:1-5:13
Regular Paper
Anna
Adamaszek
Anna Adamaszek
Antonios
Antoniadis
Antonios Antoniadis
Amit
Kumar
Amit Kumar
Tobias
Mömke
Tobias Mömke
10.4230/LIPIcs.STACS.2018.5
Anna Adamaszek, Antonios Antoniadis, and Tobias Mömke. Airports and railways: Facility location meets network design. In STACS, volume 47 of LIPIcs, pages 6:1-6:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016.
Anna Adamaszek, Artur Czumaj, Andrzej Lingas, and Jakub Onufry Wojtaszczyk. Approximation schemes for capacitated geometric network design. In Luca Aceto, Monika Henzinger, and Jirí Sgall, editors, Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part I, volume 6755 of Lecture Notes in Computer Science, pages 25-36. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22006-7_3.
http://dx.doi.org/10.1007/978-3-642-22006-7_3
Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network flows - theory, algorithms and applications. Prentice Hall, 1993.
Hyung-Chan An, Mohit Singh, and Ola Svensson. LP-based algorithms for capacitated facility location. In FOCS, pages 256-265. IEEE Computer Society, 2014.
Sanjeev Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753-782, 1998.
Babak Behsaz, Mohammad R. Salavatipour, and Zoya Svitkina. New approximation algorithms for the unsplittable capacitated facility location problem. Algorithmica, 75(1):53-83, 2016.
Raja Jothi and Balaji Raghavachari. Approximation algorithms for the capacitated minimum spanning tree problem and its variants in network design. ACM Transactions on Algorithms (TALG), 1(2):265-282, 2005.
Shi Li. A 1.488 approximation algorithm for the uncapacitated facility location problem. Inf. Comput., 222:45-58, 2013.
Jens Maßberg and Jens Vygen. Approximation algorithms for a facility location problem with service capacities. ACM Trans. Algorithms, 4(4):50:1-50:15, 2008.
R. Ravi and Amitabh Sinha. Approximation algorithms for problems combining facility location and network design. Operations Research, 54(1):73-81, 2006. URL: http://dx.doi.org/10.1287/opre.1050.0228.
http://dx.doi.org/10.1287/opre.1050.0228
Alexander Schrijver. Combinatorial Optimization. Springer, 2003.
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