Linear Programming Hierarchies Suffice for Directed Steiner Tree

  • Zachary Friggstad
  • Jochen Könemann
  • Young Kun-Ko
  • Anand Louis
  • Mohammad Shadravan
  • Madhur Tulsiani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)


We demonstrate that ℓ rounds of the Sherali-Adams hierarchy and 2ℓ rounds of the Lovász-Schrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in ℓ-layered graphs from \(\Omega(\sqrt k)\) to O(ℓ·logk) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2ℓ rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(ℓ·logk).

We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(logk) integrality gap in the standard LP relaxation, complementing the known fact that the gap can be as large as \(\Omega(\sqrt k)\) in graphs with 4 layers.


Linear Programming Relaxation Edge Cost Path Decomposition Linear Programming Solution Oracle Access 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Zachary Friggstad
    • 1
  • Jochen Könemann
    • 2
  • Young Kun-Ko
    • 3
  • Anand Louis
    • 4
  • Mohammad Shadravan
    • 2
  • Madhur Tulsiani
    • 5
  1. 1.Department of Computing ScienceUniversity of AlbertaCanada
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooCanada
  3. 3.Department of Computer SciencePrinceton UniversityUSA
  4. 4.College of ComputingGeorgia Tech.USA
  5. 5.Toyota Technical Institute at ChicagoUSA

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