Algorithmica

, Volume 30, Issue 3, pp 353–375

On Rooted Node-Connectivity Problems

  • J. Cheriyan
  • T. Jordán
  • Z. Nutov
Article

DOI: 10.1007/s00453-001-0017-7

Cite this article as:
Cheriyan, J., Jordán, T. & Nutov, Z. Algorithmica (2001) 30: 353. doi:10.1007/s00453-001-0017-7

Abstract.

Let G be a graph which is k -outconnected from a specified root node r , that is, G has k openly disjoint paths between r and v for every node v . We give necessary and sufficient conditions for the existence of a pair rv,rw of edges for which replacing these edges by a new edge vw gives a graph that is k -outconnected from r . This generalizes a theorem of Bienstock et al. on splitting off edges while preserving k -node-connectivity.

We also prove that if C is a cycle in G such that each edge in C is critical with respect to k -outconnectivity from r , then C has a node v , distinct from r , which has degree k . This result is the rooted counterpart of a theorem due to Mader.

We apply the above results to design approximation algorithms for the following problem: given a graph with nonnegative edge weights and node requirements cu for each node u , find a minimum-weight subgraph that contains max {cu,cv} openly disjoint paths between every pair of nodes u,v . For metric weights, our approximation guarantee is 3 . For uniform weights, our approximation guarantee is \min{ 2, (k+2q-1)/k} . Here k is the maximum node requirement, and q is the number of positive node requirements.

Key words. Graph connectivity, k -Connectivity, k -Outconnectivity, Splitting-off theorems, NP-hard problems, Approximation algorithms, Metric costs, Uniform costs.

Copyright information

© Springer-Verlag 2001

Authors and Affiliations

  • J. Cheriyan
    • 1
  • T. Jordán
    • 2
  • Z. Nutov
    • 3
  1. 1.Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. jcheriyan@math.uwaterloo.ca.CA
  2. 2.Department of Operations Research, Eőtvős University, 1053 Budapest, Hungary. jordan@cs.elte.hu.HU
  3. 3.Open University, Tel Aviv, Israel. nutov@oumail.openu.ac.il.IL