Polylogarithmic inapproximability

Eran Halperin*, Robert Krauthgamer

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

187 Scopus citations


We provide the first hardness result of a polylogarithmic approximation ratio for a natural NP-hard optimization problem. We show that for every fixed ε > 0, the GROUP-STEINER-TREE problem admits no efficient log2-ε k approximation, where k denotes the number of groups (or, alternatively, the input size), unless NP has quasi-polynomial Las-Vegas algorithms. This hardness result holds even for input graphs which are Hierarchically Well-Separated Trees, introduced by Bartal [FOCS, 1996]. For these trees (and also for general trees), our bound is nearly tight with the log-squared approximation currently known. Our results imply that for every fixed ε > 0, the DIRECTED-STEINER-TREE problem admits no log2-ε n-approximation, where n is the number of vertices in the graph, under the same complexity assumption.

Original languageEnglish
Pages (from-to)585-594
Number of pages10
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
StatePublished - 2003
Externally publishedYes
Event35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States
Duration: 9 Jun 200311 Jun 2003


  • Approximation algorithms
  • Hardness of approximation
  • Integrality ratio
  • Polylogarithmic approximation
  • Steiner tree


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