The Power of Recourse for Online MST and TSP

  • Nicole Megow
  • Martin Skutella
  • José Verschae
  • Andreas Wiese
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

We consider the online MST and TSP problems with recourse. The nodes of an unknown graph with metric edge cost appear one by one and must be connected in such a way that the resulting tree or tour has low cost. In the standard online setting, with irrevocable decisions, no algorithm can guarantee a constant competitive ratio. In our model we allow recourse actions by giving a limited budget of edge rearrangements per iteration. It has been an open question for more than 20 years if an online algorithm equipped with a constant (amortized) budget can guarantee constant-approximate solutions [7].

As our main result, we answer this question affirmatively in an amortized setting. We introduce an algorithm that maintains a nearly optimal tree when given constant amortized budget. In the non-amortized setting, we specify a promising proof technique and conjecture a structural property of optimal solutions that would prove a constant competitive ratio with a single recourse action. It might seem rather optimistic that such a small budget should be sufficient for a significant cost improvement. However, we can prove such power of recourse in the offline setting in which the sequence of node arrivals is known. Even this problem prohibits constant approximations if there is no recourse allowed. Surprisingly, already a smallest recourse budget significantly improves the performance guarantee from non-constant to constant.

Unlike in classical TSP variants, the standard double-tree and shortcutting approach does not give constant guarantees in the online setting. However, a non-trivial robust shortcutting technique allows to translate online MST results into TSP results at the loss of small factors.

Keywords

Span Tree Minimum Span Tree Travel Salesman Problem Travel Salesman Problem Competitive Ratio 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Nicole Megow
    • 1
  • Martin Skutella
    • 1
  • José Verschae
    • 2
  • Andreas Wiese
    • 3
  1. 1.Department of MathematicsTechnische Universität BerlinGermany
  2. 2.Departamento de Ingeniería IndustrialUniversidad de ChileChile
  3. 3.Department of Computer and System SciencesSapienza University of RomeItaly

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