Improved Approximations for TSP with Simple Precedence Constraints

(Extended Abstract)
  • Hans-Joachim Böckenhauer
  • Ralf Klasing
  • Tobias Mömke
  • Monika Steinová
Conference paper

DOI: 10.1007/978-3-642-13073-1_7

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6078)
Cite this paper as:
Böckenhauer HJ., Klasing R., Mömke T., Steinová M. (2010) Improved Approximations for TSP with Simple Precedence Constraints. In: Calamoneri T., Diaz J. (eds) Algorithms and Complexity. CIAC 2010. Lecture Notes in Computer Science, vol 6078. Springer, Berlin, Heidelberg

Abstract

In this paper, we consider variants of the traveling salesman problem with precedence constraints. We characterize hard input instances for Christofides’ algorithm and Hoogeveen’s algorithm by relating the two underlying problems, i. e., the traveling salesman problem and the problem of finding a minimum-weight Hamiltonian path between two prespecified vertices. We show that the sets of metric worst-case instances for both algorithms are disjoint in the following sense. There is an algorithm that, for any input instance, either finds a Hamiltonian tour that is significantly better than 1.5-approximative or a set of Hamiltonian paths between all pairs of endpoints, all of which are significantly better than 5/3-approximative.

In the second part of the paper, we give improved algorithms for the ordered TSP, i. e., the TSP, where the precedence constraints are such that a given subset of vertices has to be visited in some prescribed linear order. For the metric case, we present an algorithm that guarantees an approximation ratio of 2.5 − 2/k, where k is the number of ordered vertices. For near-metric input instances satisfying a β-relaxed triangle inequality, we improve the best previously known ratio to \(k\beta^{\log_2 (3k-3)}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Hans-Joachim Böckenhauer
    • 1
  • Ralf Klasing
    • 2
  • Tobias Mömke
    • 1
  • Monika Steinová
    • 1
  1. 1.Department of Computer ScienceETH ZurichSwitzerland
  2. 2.CNRS - LaBRI - Université Bordeaux 1Talence cedexFrance

Personalised recommendations