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Mathematical Programming

, Volume 160, Issue 1–2, pp 379–406 | Cite as

On the equivalence of the bidirected and hypergraphic relaxations for Steiner tree

  • Andreas Emil FeldmannEmail author
  • Jochen Könemann
  • Neil Olver
  • Laura Sanità
Full Length Paper Series A
  • 262 Downloads

Abstract

The bottleneck of the currently best \((\ln (4)+{\varepsilon })\)-approximation algorithm for the NP-hard Steiner tree problem is the solution of its large, so called hypergraphic, linear programming relaxation (HYP). Hypergraphic LPs are strongly NP-hard to solve exactly, and it is a formidable computational task to even approximate them sufficiently well. We focus on another well-studied but poorly understood LP relaxation of the problem: the bidirected cut relaxation (BCR). This LP is compact, and can therefore be solved efficiently. Its integrality gap is known to be greater than 1.16, and while this is widely conjectured to be close to the real answer, only a (trivial) upper bound of 2 is known. In this article, we give an efficient constructive proof that BCR and HYP are polyhedrally equivalent in instances that do not have an (edge-induced) claw on Steiner vertices, i.e., they do not contain a Steiner vertex with three Steiner neighbours. This implies faster \(\ln (4)\)-approximations for these graphs, and is a significant step forward from the previously known equivalence for (so called quasi-bipartite) instances in which Steiner vertices form an independent set. We complement our results by showing that even restricting to instances where Steiner vertices induce one single star, determining whether the two relaxations are equivalent is NP-hard.

Mathematics Subject Classification

90C27 68W25 

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • Andreas Emil Feldmann
    • 2
    • 3
    Email author
  • Jochen Könemann
    • 1
  • Neil Olver
    • 4
    • 5
  • Laura Sanità
    • 1
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.SZTAKIHungarian Academy of SciencesBudapestHungary
  3. 3.KAMCharles University in PraguePragueCzech Republic
  4. 4.Vrije Universiteit AmsterdamAmsterdamThe Netherlands
  5. 5.CWIAmsterdamThe Netherlands

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