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Efficient Algorithms for Airline Problem

  • Shin-ichi Nakano
  • Ryuhei Uehara
  • Takeaki Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4484)

Abstract

The airlines in the real world form small-world network. This implies that they are constructed with an ad hoc strategy. The small-world network is not so bad from the viewpoints of customers and managers. The customers can fly to any destination through a few airline hubs, and the number of airlines is not so many comparing to the number of airports. However, clearly, it is not the best solution in either viewpoint since there is a trade off. In this paper, one of the extreme cases, which is the standpoint of the manager, is considered; we assume that customers are silent and they never complain even if they are required to transit many times. This assumption is appropriate for some transportation service and packet communication. Under this assumption, the airline problem is to construct the least cost connected network for given distribution of the populations of cities with no a priori connection. First, we show an efficient algorithm that produces a good network which is minimized the number of vacant seats. The resultant network contains at most n connections (or edges), where n is the number of cities. Next we aim to minimize not only the number of vacant seats, but also the number of airline connections. The connected network with the least number of edges is a tree which has exactly n − 1 connections. However, the problem to construct a tree airline network with the minimum number of vacant seats is \({\cal NP}\)-complete. We also propose efficient approximation algorithms to construct a tree airline network with the minimum number of vacant seats.

Keywords

Airline problem approximation algorithm efficient algorithm \({\cal NP}\)-completeness 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Shin-ichi Nakano
    • 1
  • Ryuhei Uehara
    • 2
  • Takeaki Uno
    • 3
  1. 1.Department of Computer Science, Faculty of Engineering, Gunma University, Gunma 376-8515Japan
  2. 2.School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa 923-1292Japan
  3. 3.National Institute of Informatics, Hitotsubashi 2-1-2, Chiyoda-ku, Tokyo 101-8430Japan

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