Advertisement

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)

Abstract

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.

Keywords

Linear Programming Relaxation Edge Cost Path Decomposition Linear Programming Solution Oracle Access 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Calinescu, G., Zelikovsky, G.: The polymatroid Steiner problems. J. Combinatorial Optimization 9(3), 281–294 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Charikar, M., Chekuri, C., Cheung, T., Dai, Z., Goel, A., Guha, S., Li, M.: Approximation algorithms for directed Steiner problems. J. Algorithms 33(1), 73–91 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Chlamtáč, E., Tulsiani, M.: Convex relaxations and integrality gaps. In: Handbook on Semidefinite. Springer (2012)Google Scholar
  4. 4.
    Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group Steiner tree problem. J. Algorithms 37(1), 66–84 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Gupta, A., Talwar, K., Witmer, D.: Sparsest cut on bounded treewidth graphs: algorithms and hardness results. In: Proceedings of STOC (2013)Google Scholar
  6. 6.
    Guruswami, V., Sinop, A.K.: Faster SDP hierarchy solvers for local rounding algorithms. In: Proceedings of FOCS (2012)Google Scholar
  7. 7.
    Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proceedings of STOC (2003)Google Scholar
  8. 8.
    Karlin, A., Mathieu, C., Nguyen, C.: Integrlaity gaps of linear and semidefinite programming relaxations for knapsack. In: Proceedings of IPCO (2011)Google Scholar
  9. 9.
    Rothvoss, T.: Directed Steiner tree and the Lasserre hierarchy. CoRR abs/1111.5473 (2011)Google Scholar
  10. 10.
    Sherali, H., Adams, W.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3, 411–430 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Zelikovsky, A.: A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica 18, 99–110 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Zosin, L., Khuller, S.: On directed Steiner trees. In: Proceedings of SODA (2002)Google Scholar
  13. 13.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM Journal on Optimization 1, 166–190 (1991)CrossRefzbMATHMathSciNetGoogle Scholar

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

Personalised recommendations