Improved approximations for buy-at-bulk and shallow-light $$k$$ k -Steiner trees and $$(k,2)$$ ( k , 2 ) -subgraph

Abstract

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light $$k$$k-Steiner tree problem (SL$$k$$kST), we are given an undirected graph $$G=(V,E)$$G=(V,E) with terminals $$T\subseteq V$$T⊆V containing a root $$r\in T$$r?T, a cost function $$c:E\rightarrow \mathbb {R}^+$$c:E?R+, a length function $$\ell :E\rightarrow \mathbb {R}^+$$l:E?R+, a bound $$L>0$$L>0 and an integer $$k\ge 1$$k?1. The goal is to find a minimum $$c$$c-cost $$r$$r-rooted Steiner tree containing at least $$k$$k terminals whose diameter under $$\ell $$l metric is at most $$L$$L. The input to the buy-at-bulk $$k$$k-Steiner tree problem (BB$$k$$kST) is similar: graph $$G=(V,E)$$G=(V,E), terminals $$T\subseteq V$$T⊆V containing a root $$r\in T$$r?T, cost and length functions $$c,\ell :E\rightarrow \mathbb {R}^+$$c,l:E?R+, and an integer $$k\ge 1$$k?1. The goal is to find a minimum total cost $$r$$r-rooted Steiner tree $$H$$H containing at least $$k$$k terminals, where the cost of each edge $$e$$e is $$c(e)+\ell (e)\cdot f(e)$$c(e)+l(e)·f(e) where $$f(e)$$f(e) denotes the number of terminals whose path to root in $$H$$H contains edge $$e$$e. We present a bicriteria $$(O(\log ^2 n),O(\log n))$$(O(log2n),O(logn))-approximation for SL$$k$$kST: the algorithm finds a $$k$$k-Steiner tree with cost at most $$O(\log ^2 n\cdot \text{ opt }^*)$$O(log2n·opt?) where $$\text{ opt }^*$$opt? is the cost of an LP relaxation of the problem and diameter at most $$O(L\cdot \log n)$$O(L·logn). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX'06/Algorithmica'09) which had ratio $$(O(\log ^4 n), O(\log ^2 n))$$(O(log4n),O(log2n)). Using this, we obtain an $$O(\log ^3 n)$$O(log3n)-approximation for BB$$k$$kST, which improves upon the $$O(\log ^4 n)$$O(log4n)-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost $$2$$2-edge-connected subgraph with at least $$k$$k vertices, which is introduced as the $$(k,2)$$(k,2)-subgraph problem in Lau et al. (2009) (STOC'07/SICOMP09). This generalizes some well-studied classical problems such as the $$k$$k-MST and the minimum cost $$2$$2-edge-connected subgraph problems. We give an $$O(\log n)$$O(logn)-approximation algorithm for this problem which improves upon the $$O(\log ^2 n)$$O(log2n)-approximation algorithm of Lau et al. (2009).