TY - GEN

T1 - Set connectivity problems in undirected graphs and the directed Steiner network problem

AU - Chekuri, Chandra

AU - Even, Guy

AU - Gupta, Anupam

AU - Segev, Danny

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=58449134357&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58449134357&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:58449134357

SN - 9780898716474

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 532

EP - 541

BT - Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms

T2 - 19th Annual ACM-SIAM Symposium on Discrete Algorithms

Y2 - 20 January 2008 through 22 January 2008

ER -