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Approximating Fault-Tolerant Group-Steiner Problems

Authors Rohit Khandekar, Guy Kortsarz, Zeev Nutov



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Rohit Khandekar
Guy Kortsarz
Zeev Nutov

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Rohit Khandekar, Guy Kortsarz, and Zeev Nutov. Approximating Fault-Tolerant Group-Steiner Problems. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 263-274, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)
https://doi.org/10.4230/LIPIcs.FSTTCS.2009.2324

Abstract

In this paper, we initiate the study of designing approximation algorithms for {\sf Fault-Tolerant Group-Steiner} ({\sf FTGS}) problems. The motivation is to protect the well-studied group-Steiner networks from edge or vertex failures. In {\sf Fault-Tolerant Group-Steiner} problems, we are given a graph with edge- (or vertex-) costs, a root vertex, and a collection of subsets of vertices called groups. The objective is to find a minimum-cost subgraph that has two edge- (or vertex-) disjoint paths from each group to the root. We present approximation algorithms and hardness results for several variants of this basic problem, e.g., edge-costs vs. vertex-costs, edge-connectivity vs. vertex-connectivity, and $2$-connecting from each group a single vertex vs. many vertices. Main contributions of our paper include the introduction of very general structural lemmas on connectivity and a charging scheme that may find more applications in the future. Our algorithmic results are supplemented by inapproximability results, which are tight in some cases. Our algorithms employ a variety of techniques. For the edge-connectivity variant, we use a primal-dual based algorithm for covering an {\em uncros\-sable} set-family, while for the vertex-connectivity version, we prove a new graph-theoretic lemma that shows equivalence between obtaining two vertex-disjoint paths from two vertices and $2$-connecting a carefully chosen single vertex. To handle large group-sizes, we use a $p$-Steiner tree algorithm to identify the ``correct'' pair of terminals from each group to be connected to the root. We also use a non-trivial charging scheme to improve the approximation ratio for the most general problem we consider.
Keywords
  • Fault-tolerance
  • group Steiner problem
  • edge-disjointness
  • vertex-disjointness
  • approximation
  • connectivity

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