Approximating rooted connectivity augmentation problems

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

A graph is called ℓ-connected from U tor if there are ℓ internally disjoint paths from every node u ∈ U to r. The Rooted Subset Connectivity Augmentation Problem (RSCAP) is as follows: given a graph G = (V + r, E), a node subset U ⊆ V, and an integer k, find a smallest set F of new edges such that G + F is k-connected from U to r. In this paper we consider mainly a restricted version of RSCAP in which the input graph G is already (k - 1)-connected from U to r. For this version we give an O(ln |U|)-approximation algorithm, and show that the problem cannot achieve a better approximation guarantee than the Set Cover Problem (SCP) on |U| elements and with |V| - |U| sets. For the general version of RSCAP we give an O(ln k ln |U|)-approximation algorithm. For U = V we get the Rooted Connectivity Augmentation Problem (RCAP). For directed graphs RCAP is polynomially solvable, but for undirected graphs its complexity status is not known: no polynomial algorithm is known, and it is also not known to be NP-hard. For undirected graphs with the input graph G being (k - 1)-connected from V to r, we give an algorithm that computes a solution of size exceeding a lower bound of the optimum by at most (k - 1)/2 edges.

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
EditorsSanjeev Asora, Amit Sahai, Klaus Jansen, Jose D.P. Rolim
PublisherSpringer Verlag
Pages141-152
Number of pages12
ISBN (Print)3540407707, 9783540407706
DOIs
StatePublished - 2003

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume2764
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

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