Set connectivity problems in undirected graphs and the directed Steiner network problem

Chandra Chekuri, Guy Even, Anupam Gupta, Danny Segev

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In the generalized connectivity problem, we are given an edge-weighted graph G = (V,E) and a collection D = {(S 1,T 1),... ,(S k,T k)} of distinct demands; each demand (S i,T i) is a pair of disjoint vertex subsets. We say that a subgraph F ⊆ G connects a demand (S i,T i) when it contains a path with one endpoint in S i and the other in T i. The goal is to identify a minimum weight subgraph that connects all demands in D. Alon et al. (SODA '04) introduced this problem to study online network formation settings and showed that it captures some well-studied problems such as Steiner forest, non-metric facility location, tree multicast, and group Steiner tree. Finding a non-trivial approximation ratio for generalized connectivity was left as an open problem. Our starting point is the first polylogarithmic approximation for generalized connectivity, attaining a performance guarantee of O(log 2 n log 2 k). Here n is the number of vertices in G and k is the number of demands. We also prove that the cut-covering relaxation of this problem has an O(log 3 n log 2 k) integrality gap. Building upon the results for generalized connectivity, we obtain improved approximation algorithms for two problems that contain generalized connectivity as a special case. For the directed Steiner network problem, we obtain an O(k 1/2+∈) approximation, which improves on the currently best performance guarantee of Õ(k 2/3) due to Charikar et al. (SODA '98). For the set connector problem, recently introduced by Fukunaga and Nagamochi (IPCO '07), we present a polylogarithmic approximation; this result improves on the previously known ratio which can be Ω(n) in the worst case.

Original languageEnglish
Title of host publicationProceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms
Pages532-541
Number of pages10
StatePublished - 2008
Externally publishedYes
Event19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United States
Duration: 20 Jan 200822 Jan 2008

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference19th Annual ACM-SIAM Symposium on Discrete Algorithms
Country/TerritoryUnited States
CitySan Francisco, CA
Period20/01/0822/01/08

ASJC Scopus subject areas

  • Software
  • Mathematics (all)

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