Improved Approximations for Buy-at-Bulk and Shallow-Light k-Steiner Trees and (k,2)-Subgraph

Abstract

In this paper we give improved approximation algorithms for some network design problems. In the Bounded-Diameter or Shallow-Light k-Steiner tree problem (SLkST), we are given an undirected graph G=(V,E) with terminals T⊆V containing a root r∈T, a cost function c:E→ℝ+, a length function l:E→ℝ+, a bound L>0 and an integer k≥1. The goal is to find a minimum c-cost r-rooted Steiner tree containing at least k terminals whose diameter under l metric is at most L. The input to the Buy-at-Bulk k-Steiner tree problem (BBkST) is similar: graph G=(V,E), terminals T⊆V, cost and length functions c,l:E→ℝ+, and an integer k≥1. The goal is to find a minimum total cost r-rooted Steiner tree H containing at least k terminals, where the cost of each edge e is c(e)+l(e)·f(e) where f(e) denotes the number of terminals whose path to root in H contains edge e. We present a bicriteria (O(log2n),O(logn))-approximation for SLkST: the algorithm finds a k-Steiner tree of diameter at most O(L·logn) whose cost is at most $O(\log^2 n\cdot\mbox{\sc opt}^*)$ where $\mbox{\sc opt}^*$ is the cost of an LP relaxation of the problem. This improves on the algorithm of [9] with ratio (O(log4n), O(log2n)). Using this, we obtain an O(log3n)-approximation for BBkST, which improves upon the O(log4n)-approximation of [9]. We also consider the problem of finding a minimum cost 2-edge-connected subgraph with at least k vertices, which is introduced as the (k,2)-subgraph problem in [14]. We give an O(logn)-approximation algorithm for this problem which improves upon the O(log2n)-approximation of [14].