Journal of Combinatorial Optimization

, Volume 31, Issue 2, pp 669–685

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

• M. Reza Khani
Article

## 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 (SL$$k$$ST), we are given an undirected graph $$G=(V,E)$$ with terminals $$T\subseteq V$$ containing a root $$r\in T$$, a cost function $$c:E\rightarrow \mathbb {R}^+$$, a length function $$\ell :E\rightarrow \mathbb {R}^+$$, a bound $$L>0$$ and an integer $$k\ge 1$$. The goal is to find a minimum $$c$$-cost $$r$$-rooted Steiner tree containing at least $$k$$ terminals whose diameter under $$\ell$$ 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 $$T\subseteq V$$ containing a root $$r\in T$$, cost and length functions $$c,\ell :E\rightarrow \mathbb {R}^+$$, and an integer $$k\ge 1$$. 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 $$c(e)+\ell (e)\cdot f(e)$$ 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 $$O(\log ^2 n\cdot \text{ opt }^*)$$ where $$\text{ opt }^*$$ is the cost of an LP relaxation of the problem and diameter at most $$O(L\cdot \log n)$$. 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).

## Keywords

Combinatorial optimization Approximation algorithms Network design Steiner tree $$k$$-edge connected

## Notes

### Acknowledgments

We would like to thank an anonymous referee for her/his careful reading of this paper and the comments and suggestions.

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