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 (SL k ST), 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 (BB k ST) is similar: graph G = ( V , E ) , terminals T ⊆ V containing a root r ∈ T , 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 ( log 2 n ) , O ( log n ) ) -approximation for SL k ST: the algorithm finds a k -Steiner tree with cost at most O ( log 2 n · opt ∗ ) where opt ∗ is the cost of an LP relaxation of the problem and diameter at most O ( L · log n ) . This improves on the algorithm of Hajiaghayi et al. ( 2009 ) (APPROX’06/Algorithmica’09) which had ratio ( O ( log 4 n ) , O ( log 2 n ) ) . Using this, we obtain an O ( log 3 n ) -approximation for BB k ST, which improves upon the O ( log 4 n ) -approximation of Hajiaghayi et al. ( 2009 ). 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 Lau et al. ( 2009 ) (STOC’07/SICOMP09). This generalizes some well-studied 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 2 n ) -approximation algorithm of Lau et al. ( 2009 ).