ISAAC 2011: Algorithms and Computation pp 20-29

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

• M. Reza Khani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)

## 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(ef(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].

## Keywords

Approximation Algorithm Steiner Tree Length Function Network Design Problem Minimum Total Cost
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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