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A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs

Authors Zachary Friggstad, Jochen Könemann, Mohammad Shadravan

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Zachary Friggstad
Jochen Könemann
Mohammad Shadravan

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Zachary Friggstad, Jochen Könemann, and Mohammad Shadravan. A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 3:1-3:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SWAT.2016.3

Abstract

We demonstrate that the integrality gap of the natural cut-based LP relaxation for the directed Steiner tree problem is O(log k) in quasi-bipartite graphs with k terminals. Such instances can be seen to generalize set cover, so the integrality gap analysis is tight up to a constant factor.  A novel aspect of our approach is that we use the primal-dual method; a technique that is rarely used in designing approximation algorithms for network design problems in directed graphs.

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Keywords
  • Approximation algorithm
  • Primal-Dual algorithm
  • Directed Steiner tree

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