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(log² n),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(log⁴ n), O(log² n)). Using this, we obtain an O(log³ n)-approximation for BBkST, which improves upon the O(log⁴ n)-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(log² n)-approximation of [14].