Fourth IFIP International Conference on Theoretical Computer Science- TCS 2006
Volume 209 of the series IFIP International Federation for Information Processing pp 251-270
Reusing Optimal TSP Solutions for Locally Modified Input Instances
- Hans-Joachim BöckenhauerAffiliated withDepartment of Computer Science, ETH Zurich
- , Luca ForlizziAffiliated withDepartment of Computer Science, Università di L’Aquila
- , Juraj HromkovičAffiliated withDepartment of Computer Science, ETH Zurich
- , Joachim KneisAffiliated withDepartment of Computer Science, RWTH Aachen University
- , Joachim KupkeAffiliated withDepartment of Computer Science, ETH Zurich
- , Guido ProiettiAffiliated withDepartment of Computer Science, Università di L’AquilaIstituto di Analisi dei Sistemi ed Informatiea “A. Ruberti”, CNR
- , Peter WidmayerAffiliated withDepartment of Computer Science, ETH Zurich
Abstract
- 1.
The local modification to change the cost of a singular edge turns the traveling salesperson problem (TSP) into a problem LM-TSP which is as hard as TSP itself, i.e., unless P=NP, there is no polynomial-time p(n)-approximation algorithm for LM-TSP for any polynomial p. Moreover, LM-TSP where inputs must satisfy the β triangle inequality (LM-Δ β -TSP) remains NP-hard for all β > 1/2.
- 2.
For LM-Δ-TSP (i.e., metric LM-TSP), an efficient 1.4-approximation algorithm is presented. In other words, the additional information enables us to do better than if we simply used Christofides’ algorithm for the modified input.
- 3.
Similarly, for all 1 < β < 3.34899, we achieve a better approximation ratio for LM-Δ β -TSP than for Δ’-TSP.
- 4.
Metric TSP with deadlines (time windows), if a single deadline or the cost of a single edge is modified, exhibits the same lower bounds on the approximability in these local-modification versions as those currently known for the original problem. instance. A second construction inflates this advantage. Tours which start at time X, different from those that start between times X+g and X +ςg, may spend some extra time to visit a group of vertices which, unless visited early, will cause belated tours to run k times zigzag across a huge distance γ.
The following lemma describes the construction in detail. See Figure 5 for an overview.
- Title
- Reusing Optimal TSP Solutions for Locally Modified Input Instances
- Book Title
- Fourth IFIP International Conference on Theoretical Computer Science- TCS 2006
- Book Subtitle
- IFIP 19th Worm Computer Congress, TC-1, Foundations of Computer Science, August 23–24, 2006, Santiago, Chile
- Book Part
- Part III
- Pages
- pp 251-270
- Copyright
- 2006
- DOI
- 10.1007/978-0-387-34735-6_21
- Print ISBN
- 978-0-387-34633-5
- Online ISBN
- 978-0-387-34735-6
- Series Title
- IFIP International Federation for Information Processing
- Series Volume
- 209
- Series ISSN
- 1571-5736
- Publisher
- Springer US
- Copyright Holder
- International Federation for Information Processing
- Additional Links
- Topics
- Industry Sectors
- eBook Packages
- Editors
-
- Gonzalo Navarro (1)
- Leopoldo Bertossi (2)
- Yoshiharu Kohayakawa (3)
- Editor Affiliations
-
- 1. Universidad de Chile
- 2. Carleton University
- 3. Universidade de Sao Paulo
- Authors
-
- Hans-Joachim Böckenhauer (4)
- Luca Forlizzi (5)
- Juraj Hromkovič (4)
- Joachim Kneis (6)
- Joachim Kupke (4)
- Guido Proietti (5) (7)
- Peter Widmayer (4)
- Author Affiliations
-
- 4. Department of Computer Science, ETH Zurich, Switzerland
- 5. Department of Computer Science, Università di L’Aquila, Italy
- 6. Department of Computer Science, RWTH Aachen University, Germany
- 7. Istituto di Analisi dei Sistemi ed Informatiea “A. Ruberti”, CNR, Roma, Italy
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