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

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 Steinter 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 treeH 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(log n), O(log n))approximation for SLkST: the algorithm finds a k-Steiner tree of diameter at most O(L · logn) whose cost is at most O(log n · opt∗) where opt∗ is the cost of an LP relaxation of the problem. This improves on the algorithm of [25] (APPROX’06/Algorithmica’09) which had ratio (O(log n), O(log n)). Using this, we obtain an O(log n)-approximation for BBkST, which improves upon the O(log n)-approximation of [25]. 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 [32] (STOC’07/SICOMP09). This generalizes some wellstudied classical problems such as the k-MST and the minimum cost 2-edge-connected subgraph problems. We give an O(log n)-approximation algorithm for this problem which improves upon the O(log n)-approximation of [32]