On-line generalized Steiner problem

Abstract

The generalized Steiner problem (GSP) is defined as follows. We are given a graph with non-negative edge weights and a set of pairs of vertices. The algorithm has to construct minimum weight subgraph such that the two nodes of each pair are connected by a path. Off-line GSP approximation algorithms were given in Agarwal et al. (SIAM J. Comput. 24(3) (1995) 440) and Goemans and Williamson (SIAM J. Comput. 24(2) (1995) 296). We consider the on-line GSP, in which pairs of vertices arrive on-line and are needed to be connected immediately. We show that the online Min-Cost (i.e. greedy) strategy for this problem has O( log 2 n) competitive ratio. The previous best algorithm was O( n log n) competitive (Workshop on Algorithms and Data Structures, 1993, pp. 622–633). Following this work a different (non-greedy) algorithm has been shown to achieve an O( log n) competitive ratio (Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 344–353). We also consider the network connectivity leasing problem which is a generalization of the GSP. Here, edges of the graph can be either bought or leased for different costs. We provide simple randomized algorithm based on on-line generalized Steiner algorithms whose competitive ratio is within a constant factor of the best competitive algorithm for the on-line GSP.